As a matter of fact, you CAN enlarge the root filesystem while Ubuntu is running (I learned this recently myself here) - this sounds incredible but it's true :)

Here's the list of steps for a simple scenario where you have two partitions, /dev/sda1 is an ext4 partition the OS is booted from and /dev/sdb2 is swap. For this exercise we want to remove the swap partition an extend /dev/sda1 to the whole disk.

1. As always, make sure you have a backup of your data - since we're going to modify the partition table there's a chance to lose all your data if you make a typo, for example.

2. Run sudo fdisk /dev/sda

   • use p to list the partitions. Make note of the start cylinder of /dev/sda1
   • use d to delete first the swap partition (2) and then the /dev/sda1 partition. This is very scary but is actually harmless as the data is not written to the disk until you write the changes to the disk.
   • use n to create a new primary partition. Make sure its start cylinder is exactly the same as the old /dev/sda1 used to have. For the end cylinder agree with the default choice, which is to make the partition to span the whole disk.
   • use a to toggle the bootable flag on the new /dev/sda1
   • review your changes, make a deep breath and use w to write the new partition table to disk. You'll get a message telling that the kernel couldn't re-read the partition table because the device is busy, but that's ok.

3. Reboot with sudo reboot. When the system boots, you'll have a smaller filesystem living inside a larger partition.

4. The next magic command is resize2fs. Run sudo resize2fs /dev/sda1 - this form will default to making the filesystem to take all available space on the partition.

That's it, we've just resized a partition on which Ubuntu is installed, without booting from an external drive.

Two Methods

1. Tested in Ubuntu Server 16.04 to 18.10

After extending the volume in VMware ESXi (shutdown vm, backup/export vm, remove all snapshots, then increased number in disk size, take snapshot of vm again before following the next steps so you can revert mistake/error).

The following example is extending volume on root partition on a system that is already using LVM.

1. ssh into the system, switch to root and run cfdisk
   # cfdisk

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. After exiting cfdisk

     # fdisk -l /dev/sda 
     Disk /dev/sda: 20 GiB, 21474836480 bytes, 41943040 sectors
     Units: sectors of 1 * 512 = 512 bytes
     Sector size (logical/physical): 512 bytes / 512 bytes
     I/O size (minimum/optimal): 512 bytes / 512 bytes
     Disklabel type: dos
     Disk identifier: 0xc8b647ff
    
     Device     Boot    Start      End  Sectors  Size Id Type
     /dev/sda1  *        2048   999423   997376  487M 83 Linux
     /dev/sda2        1001470 16775167 15773698  7.5G  5 Extended
     /dev/sda3       16775168 41943039 25167872   12G 8e Linux LVM (***** See new partition /dev/sda3*****)
     /dev/sda5        1001472 16775167 15773696  7.5G 8e Linux LVM
    

14. Reboot system before next step (or else you might get Device /dev/sda3 not found (or ignored by filtering))

     # shutdown now -r
    

15. Initialize the new volume

     # pvcreate /dev/sda3
       Physical volume "/dev/sda3" successfully created
    

16. Get the name of volume to extend

     # vgdisplay
       --- Volume group ---
       VG Name               linuxetc1-vg (***** this is vg i'm extending*****)
       System ID             
       Format                lvm2
       Metadata Areas        1
       Metadata Sequence No  3
       VG Access             read/write
       VG Status             resizable
       MAX LV                0
       Cur LV                2
       Open LV               2
       Max PV                0
       Cur PV                1
       Act PV                1
       VG Size               7.52 GiB
       PE Size               4.00 MiB
       Total PE              1925
       Alloc PE / Size       1925 / 7.52 GiB
       Free  PE / Size       0 / 0  
       ...
    

17. Extend volume group

     # vgextend linuxetc1-vg /dev/sda3
       Volume group "linuxetc1-vg" successfully extended
    

18. Extend root

     # lvextend -l+100%FREE /dev/linuxetc1-vg/root
       Size of logical volume linuxetc1-vg/root changed from 6.52 GiB (1669 extents) to 18.52 GiB (4741 extents).
       Logical volume root successfully resized.
    
    
     Or if you only wanted to give 5 more GB to root
     # lvextend -L +5g /dev/linuxetc1-vg/root
    

19. Get filesystem name for next step

     # df -h
     Filesystem                      Size  Used Avail Use% Mounted on
     udev                            477M     0  477M   0% /dev
     tmpfs                           100M  4.6M   95M   5% /run
     /dev/mapper/linuxetc1--vg-root  6.3G  2.5G  3.5G  42% /            (***** this line*****)
     tmpfs                           497M     0  497M   0% /dev/shm
     tmpfs                           5.0M     0  5.0M   0% /run/lock
     tmpfs                           497M     0  497M   0% /sys/fs/cgroup
     /dev/sda1                       472M  105M  343M  24% /boot
     tmpfs                           100M     0  100M   0% /run/user/1000
    

20. Extend filesystem

     # resize2fs /dev/mapper/linuxetc1--vg-root
     resize2fs 1.42.13 (17-May-2015)
     Filesystem at /dev/mapper/linuxetc1--vg-root is mounted on /; on-line resizing required
     old_desc_blocks = 1, new_desc_blocks = 2
     The filesystem on /dev/mapper/linuxetc1--vg-root is now 4854784 (4k) blocks long.
    

21. See the increased size

     # df -h
     Filesystem                      Size  Used Avail Use% Mounted on
     udev                            477M     0  477M   0% /dev
     tmpfs                           100M  4.6M   95M   5% /run
     /dev/mapper/linuxetc1--vg-root   19G  2.5G   15G  15% /         (***** this line*****)
     tmpfs                           497M     0  497M   0% /dev/shm
     tmpfs                           5.0M     0  5.0M   0% /run/lock
     tmpfs                           497M     0  497M   0% /sys/fs/cgroup
     /dev/sda1                       472M  105M  343M  24% /boot
     tmpfs                           100M     0  100M   0% /run/user/1000
    

2. Faster Method Tested in Ubuntu Server 18.04.3 LTS

This method is a little faster than the first. This is due to a new option in the cfdisk menu called "Resize" that saves some steps.

After extending the volume in VMware ESXi (shutdown vm, backup/export vm, remove all snapshots, then increased number in disk size, take snapshot of vm again before following the next steps so you can revert mistake/error).

The following example is extending volume on root partition on a system that is already using LVM.

1. ssh into the system, switch to root and run cfdisk.

2. Choose the partition to extend and select "Resize".

3. Set the "New size".

4. After pressing enter, you'll see screen with the following note "Partition [someNumber] resized":

5. Next you'll need to "Write" (save) your changes:

6. Quit cfdisk. When you exit you may see message "syncing disks".

7. I rebooted my system at this point but it may not be necessary. Now you can pickup from step 19 from the first method (repeated below).

8. (19) Get filesystem name for next step

    # df -h
    Filesystem                      Size  Used Avail Use% Mounted on
    udev                            477M     0  477M   0% /dev
    tmpfs                           100M  4.6M   95M   5% /run
    /dev/mapper/linuxetc1--vg-root  6.3G  2.5G  3.5G  42% /            (***** this line*****)
    tmpfs                           497M     0  497M   0% /dev/shm
    tmpfs                           5.0M     0  5.0M   0% /run/lock
    tmpfs                           497M     0  497M   0% /sys/fs/cgroup
    /dev/sda1                       472M  105M  343M  24% /boot
    tmpfs                           100M     0  100M   0% /run/user/1000
   

9. (20) Extend filesystem

    # resize2fs /dev/mapper/linuxetc1--vg-root
    resize2fs 1.42.13 (17-May-2015)
    Filesystem at /dev/mapper/linuxetc1--vg-root is mounted on /; on-line resizing required
    old_desc_blocks = 1, new_desc_blocks = 2
    The filesystem on /dev/mapper/linuxetc1--vg-root is now 4854784 (4k) blocks long.
   

10. (21) See the increased size

     # df -h
     Filesystem                      Size  Used Avail Use% Mounted on
     udev                            477M     0  477M   0% /dev
     tmpfs                           100M  4.6M   95M   5% /run
     /dev/mapper/linuxetc1--vg-root   19G  2.5G   15G  15% /         (***** this line*****)
     tmpfs                           497M     0  497M   0% /dev/shm
     tmpfs                           5.0M     0  5.0M   0% /run/lock
     tmpfs                           497M     0  497M   0% /sys/fs/cgroup
     /dev/sda1                       472M  105M  343M  24% /boot
     tmpfs                           100M     0  100M   0% /run/user/1000
    

Articles:
https://wiki.ubuntu.com/Lvm
http://www.geoffstratton.com/expand-hard-disk-ubuntu-lvm

EDIT: As mentioned below by @kent this is an old answer and no longer true.

You are almost there.
You can not resize the partition using GParted while Ubuntu is running.
You will need to attach the GParted ISO's as a CD to the VM machine and reboot the machine so that the GParted will be loaded instead of Ubuntu (I think you can boot from the virtual CD by pressing F12 immediately after machine is started).
Once you booted into GParted the option to move/resize will be enabled as Ubuntu is not currently running.

jtlindsey's "faster" method works on LVMs but is missing some steps. df -h wouldn't show the additional space.

1. Follow up until step 9.
2. Run sudo lsblk to determine your partition name (mine was sda3).
3. Run sudo pvresize /dev/sda3
4. Run sudo lvextend -L +76G /dev/mapper/ubuntu--vg-ubuntu--lv (replacing the size and the name with your own)
5. Go back to his method and finish remaining steps.

Source: https://askubuntu.com/a/1047512/393368

I noticed that my ubuntu VM had a LVM that simply was not using all the allocated space. So to add space to my LVM it was quite simply this: https://wiki.ubuntu.com/Lvm chapter: "resizing partitions".

For example: A quick way to know if you have an LVM is to do df -h and look for this line:

/dev/mapper/ubuntu--vg-ubuntu--lv  689G   20G  639G   3% /

And... see that is not actually the classic "/dev/sda1" type of mount point.

Then run the extension commands on that partition.

sudo lvextend -L +500g /dev/mapper/ubuntu--vg-ubuntu--lv
sudo resize2fs /dev/mapper/ubuntu--vg-ubuntu--lv

Another quick clue that reminded me to check for an LVM was that I noticed when I looked at the fdisk partitions and saw that I had a large /dev/sda3 partition that was not really mounted anywhere. (ie: when I checked with the mount command).

The answer from Sergey was excellent. If you have the empty space not directly in the next device, however, you will need to delete other devices. For example:

Device     Boot    Start       End  Sectors Size Id Type
/dev/sda1  *        2048  33556477 33554430  16G 83 Linux
/dev/sda2       33556478  41940991  8384514   4G  5 Extended
/dev/sda3       41940992 125829119 83888128  40G 83 Linux
/dev/sda5       33556480  41940991  8384512   4G 82 Linux swap / Solaris

In the above case, if you want to merge /dev/sda3 with /dev/sda1, you would need to remove /dev/sda1, /dev/sda2, and /dev/sda3 (with d command), and re-add /dev/sda2 and update its type (with n and t commands).

I succeeded in enlarging my primary ext4 partition using the methodology described here.

Some notes from my personal experience:

• The link example is for shrinking partition, hence the order is resize2fs first and then fdisk. When enlarging partition (as in my case) the order should be fdisk first, reboot and then resize2fs. Also, when using fdisk, the primary as well as the extended partitions should be removed and re-defined.
• I booted the Ubuntu from another drive and then worked on this drive which actually was mirror (using dd) of the older drive. Made it bootable using GParted
• The current version of resize2fs (I am using Ubuntu 12.04LTS) support ext4 - no need to do the actions described in the link of changing it the ext2.

With cfdisk and resize2fs I was able to extend my / partition on proxmox VM. Thanks to jtlindsey for the example.

I had the next picture before extending:

Model: QEMU QEMU HARDDISK (scsi)
Disk /dev/sda: 107GB
Sector size (logical/physical): 512B/512B
Partition Table: linux
Disk Flags: 

Number  Start   End     Size    Type      File system     Flags
 1      1049kB  40,0GB  40,0GB  primary   ext4            boot
 2      40,0GB  53,7GB  13,7GB  extended
 5      40,0GB  53,7GB  13,7GB  logical   linux-swap(v1)

With cfdisk I've removed swap partition, then resized my / partition with Resize option to 80Gb and recreated the swap partition. Then I've written changes with Write option:

    Device               Boot                         Start                 End             Sectors          Size          Id Type
>>  /dev/sda1            *                             2048           167774207           167772160           80G          83 Linux                        
    /dev/sda2                                     167774208           209715199            41940992           20G          82 Linux swap / Solaris


┌───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│Partition type: Linux swap / Solaris (82)                                                                                                              │
└───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘
                              [Bootable]  [ Delete ]  [ Resize ]  [  Quit  ]  [  Type  ]  [  Help  ]  [  Write ]  [  Dump  ]

After this I've rebooted and used the following command to extend my file system to the current partition's size:

$ sudo resize2fs /dev/sda1

Make sure you've created backup of important data before any disk changing action. So, jtlindsey's method could be used even for none-LVM partitions.

Also do not forget to get swap back by:

$ sudo mkswap /dev/sda2

And change UUID of swap partition in /etc/fstab to what sudo blkid shows for it.