The simplest set of instructions I always used for kernel upgrade / downgrade are by ubuntuforums.org user by the name of lykwydchykyn (url modified by me for this post):

1. Go here: http://kernel.ubuntu.com/~kernel-ppa/mainline/

2. Download 3 (maybe 4) debs to a folder somewhere:

   linux-headers-VERSION-NUMBER_all.deb
   linux-headers-VERSION-NUMBER_amd64.deb
   linux-image-VERSION-NUMBER_amd64.deb
   linux-image-extra-VERSION-NUMBER_amd64.deb   # if available
   

3. Install the debs with whatever package manager front-end you use (is gdebi still around?), or use these commands:

   cd /path/to/folder/where/you/put/the/debs
   sudo dpkg -i *.deb
   

Sources:

• https://wiki.ubuntu.com/Kernel/MainlineBuilds
• http://ubuntuforums.org/showthread.php?p=11391743#post11391743

You could always do the following:

apt-cache search linux-image

Pick the one you want and then do:

sudo apt-get install linux-image-your_version_choice linux-headers-your_version_choice linux-modules-extra-your_version_choice

My answer is, YES you can. The stable release was 3.4, but in this tutorial i use 3.3.1.

Ubuntu (32-bit) Generic PAE:

Run the following commands:

mkdir kernel\ v3.3.1-precise && cd kernel\ v3.3.1-precise
wget http://kernel.ubuntu.com/~kernel-ppa/mainline/v3.3.1-precise/linux-headers-3.3.1-030301-generic-pae_3.3.1-030301.201204021435_i386.deb
wget http://kernel.ubuntu.com/~kernel-ppa/mainline/v3.3.1-precise/linux-headers-3.3.1-030301_3.3.1-030301.201204021435_all.deb
wget http://kernel.ubuntu.com/~kernel-ppa/mainline/v3.3.1-precise/linux-image-3.3.1-030301-generic-pae_3.3.1-030301.201204021435_i386.deb
sudo dpkg -i linux-*.deb 
sudo update-grub
sudo reboot now

Ubuntu (64-bit):

Run the following commands:

mkdir kernel\ v3.3.1-precise && cd kernel\ v3.3.1-precise
wget http://kernel.ubuntu.com/~kernel-ppa/mainline/v3.3.1-precise/linux-headers-3.3.1-030301-generic_3.3.1-030301.201204021435_amd64.deb
wget http://kernel.ubuntu.com/~kernel-ppa/mainline/v3.3.1-precise/linux-headers-3.3.1-030301_3.3.1-030301.201204021435_all.deb
wget http://kernel.ubuntu.com/~kernel-ppa/mainline/v3.3.1-precise/linux-image-3.3.1-030301-generic_3.3.1-030301.201204021435_amd64.deb
sudo dpkg -i linux-*.deb 
sudo update-grub
sudo reboot now

Enjoy!

Contrary to some answers here Ubuntu releases the new kernel within a day or two of the main kernel team. I used to upgrade the hard way like some of the answers here suggest but I found this is an easier way.

Implications of manually installing Kernels

Manually installing kernels requires extra work.

There are reasons why you want to install the latest mainline kernel:

• A bug in the last Ubuntu LTS kernel update and you can't downgrade
• You have new hardware not supported in the current Ubuntu LTS kernel update stream or HWE updates
• You want a security upgrade or new feature only available in the latest mainline kernel version.

As of January 15, 2018 the latest stable mainline kernel is 4.14.13. At the time of writing many are interested to install it for protection against Meltdown security hole. If you choose to manually install it you should know:

• Older LTS kernels will not get updated until they are greater than the main menu first option titled Ubuntu.
• Manually installed kernels are not removed with the usual sudo apt auto-remove command. You need to follow this: How do I remove old kernel versions to clean up the boot menu?
• Monitor developments in the older kernels for when you want to get back on the regular LTS kernel update method. Then delete the manually installed mainline kernel as described in the previous bullet point link.
• After manually removing the newest mainline kernel run sudo update-grub and then Ubuntu's latest LTS kernel will be the first option called Ubuntu on Grub's main menu.

Finding the latest kernel

Go to (kernel.ubuntu.com - Kernel PPA Mainline) and press the End key:

Now click on the link 4.9.8 (or a newer one if available when you read this) and the following appears:

Note the links with the black airbrush marks. These are the three we will download for Ubuntu 16.04 64-Bit using Intel or AMD processors.

Installing using Terminal

Open a terminal session with Ctrl+Alt+T and use:

cd ~/Downloads # Change to your downloads directory
ll linux*.deb  # Check if any previous downloads still exist
rm linux*.deb  # Use this command if any previous downloads found

Now switch back to the download screen above and single-click on all three of the .deb kernel downloads. After downloads complete (you'll see status in your browser) switch back to terminal session and use:

sudo dpkg -i linux*.deb
rm linux*.deb
sudo reboot

Voila! - you are running the latest kernel when the first option in grub boot menu is selected. Your older kernel versions are still available under grub's Advanced Options menu.

After installing the newest kernel a dozen times you will ask the question "How do I remove older kernel versions?" and you will find those answers here in Ask Ubuntu.

Late but new answer.

I will add some more information in my answer. Now my old kernel was linux-headers-4.8.0-53. To find out which kernel you have type uname -r in terminal. Now type this command to get list of available kernels.

apt-cache search linux-image

This will give you a very big output. Now chose one of them that you want to install and install it by following commands(Don't type these commands like this, replace the words kernal_version with the kernel version which you wish to install)

sudo apt-get install linux-image-kernal_version
sudo apt-get install linux-image-extra-kernal_version
sudo apt-get install linux-headers-kernal_version

Now in my case I choose linux-image-4.10.0-22-generic so my command will look like this

sudo apt-get install linux-image-4.10.0-22-generic 
sudo apt-get install linux-image-extra-4.10.0-22-generic
sudo apt-get install linux-headers-4.10.0-22-generic

Now in the end remove old kernel by command

sudo apt-get autoremove

Now when I first did it, I Don't know is it my hardware problem or because of removing the old kernel, but my laptop shows me black screen which happens to my laptop sometimes. Thus I remove it's battery and rebooted and typed the command. Also I don't suffer from my black screen issue any more.

uname -r

and it gives me

linux-image-4.10.0-22-generic

which means kernel is updated.

There is a much simpler way of doing this, and it's the way I use. Simply download the Ubuntu Mainline Kernel Updater script and run it from a terminal. It will download the .deb's for you, install the new kernel, and optionally remove the old one (not recommended) then optionally reboot.

Example output (copied from my terminal):

evilsupahfly@the-black-beast:~$ sudo KernelUpdateChecker -no-rc -r utopic
Run this command to install the new kernel
/tmp/kernel-update
root@the-black-beast:~# /tmp/kernel-update
Config Notes:
Rejecting Release Candidates
Accepting Latest Kernel
Accepting kernels compiled for utopic
Accepting kernels with a version higher than 3.15.0-031500-lowlatency

Information:
Origin: 
    http://kernel.ubuntu.com/~kernel-ppa/mainline/v3.15.1-utopic
Kernel Version:
    3.15.1-031501
Release Date:
    2014/06/16 @ 18:41 (YYYY/MM/DD @ HH:MM)

Care to look at the change log? (y=Yes, n=No) (n)

(download output skipped)    

Greetings root
    I (the-black-beast) am now self aware and a sentient being, have a nice day.
    Oh and by the way SKYNET is now active.

Installing Linux 3.15.1-031501:

(installer output skipped)

The New Kernel looks to have been installed

WARNING: If the new kernel does not boot you may regret saying yes here.
Would you like to remove the current one? (y=Yes, n=No) (n): n

Are you ready to Reboot? (y=Yes, n=No) (n): n

Almost entirely automated, and completely problem free.

Run the following Terminal commands to install a new Ubuntu mainline kernel.

sudo apt-get update
sudo apt-get install python-bs4 python-apt
cd /tmp
rm -rf medigeek-kmp*
wget https://github.com/medigeek/kmp-downloader/tarball/master -O kmpd.tar.gz
tar xzf kmpd.tar.gz
cd medigeek-*
python kmpd.py -d

The script is hosted at Github. Just press Enter instead of a number if you get stuck on a certain question in the Python script.

I wrote a script and always will be up to date.

ukupgrade: Ubuntu Kernel Upgrade

• Open your favorite terminal and run the follwing commands

• Give executable permission to file

  chmod +x ./ukupgrade

• Call the script

  ./ukupgrade

If you don't want to go through the hassle to manually upgrade the kernel you can try UUKU. Works fine even on Ubuntu derivatives like elementory, mint.

I used it to fix my WiFi issue as sometimes some new hardware won't be detected by old kernel. Other than hardware driver update I don't find the need to upgrade kernel. New kernels do have security and slight performance upgrade but also have tendency to break things like your VM.

sudo apt-add-repository -y ppa:teejee2008/ppa
sudo apt-get update
sudo apt-get install ukuu
ukuu-gtk

SOURCE:

http://www.makeuseof.com/tag/upgrade-kernel-ukuu-ubuntu/

http://www.teejeetech.in/2016/07/ukuu-v167.html?m=1

Another option is to try customized and optimized builds, such as this i3/i5/i7 optimized 3.2.1 kernel for Ubuntu:

DuoPetalFlower, My Experiments with Linux - 3.2.1 kernel

He also has Intel atom optimized builds which can work quite well if you're trying to squeeze every last ounce of performance out of a netbook.

There are other kernels like Liquorix that claim to be better optimized for desktop performance.

Though not recommended by some, you can often run official kernels from later versions of Ubuntu without issues. I'm currently running the 3.3.3 precise kernel on oneiric and my machine works better than ever. Performance & battery life increased, while temperatures dropped a few degrees. My issue of a black screen when resuming from standby has also disappeared. Though I personally haven't had issues from using newer kernels, you will find some people who are strongly opposed to doing this.