You can use the Libreoffice 4.4 ppa from the libreoffice packaging team, which offers stable backports for Precise (12.04), Trusty (14.04), Utopic (14.10) and Vivid (15.04).

Note: if you rely on libreoffice for work or school you may not want to risk updating, even though the backports provided are largely stable.

sudo add-apt-repository ppa:libreoffice/libreoffice-4-4
sudo apt-get update
sudo apt-get dist-upgrade

Obviously, if you don't already have libreoffice installed, you will also need to run

sudo apt-get install libreoffice

It should be noted that this ppa will only produce updates for the 4.4 series, while the main libreoffice ppa will provide updates beyond the 4.4 series, although that ppa is largely for testing and development builds.

(The original 4.0 series ppa is here at launchpad, if anyone has any special reason to use that series.)

For more information, see this very recent useful article on libreoffice 4, and the getting started guide at the official documentation site.

This method is not recommended!

While using this method will still work fine, and you will end up with the latest version of LibreOffice, LibreOffice 4.2 is now available in the Trusty repositories and the recommended method of installation is using the software center or apt.

If you want to live on the wild side, or you want the absolute latest and greatest version of LibreOffice use this answer.

If you find something out of date, please leave a comment below or go ahead and fix it yourself :)

Start by downloading it:

• Direct torrent link: LibreOffice 4.3.4 torrent 64 bit or LibreOffice 4.3.4 torrent 32 bit

• Normal installer direct link: LibreOffice 4.3.4 normal installer 64 bit or LibreOffice 4.3.0 normal installer 32 bit

I recommend the torrent for faster download speed and for slow connections.

To install LibreOffice 4 you will need to remove all previous versions. Run:

sudo apt-get remove --purge libreoffice-core libreoffice-common
sudo apt-get autoremove --purge

Note: I'm not 100% sure this will remove all of LibreOffice. If you experience any errors later, please let me know.

Extract the files:

1. cd to the Downloads directory:

   cd Downloads  
   

2. Extract the tar.gz:

   For 64 bit:

   tar -xvzf LibreOffice_4.3.4_Linux_x86-64_deb.tar.gz  
   

   For 32 bit:

   tar -xvzf LibreOffice_4.3.4_Linux_x86_deb.tar.gz
   

Install the program:

1. cd to the programs folder:

   For 64 bit:

   cd LibreOffice_4.3.4_Linux_x86-64_deb/DEBS  
   

   For 32 bit:

   cd LibreOffice_4.3.4_Linux_x86_deb/DEBS
   

2. Install part one (for both 32 and 64 bit):

   sudo dpkg -i *.deb
   

You're done! You can now remove both the directory and the tarball:

rm -r ~/Downloads/LibreOffice_4.3.4_Linux_x86-64_deb 
rm ~/Downloads/LibreOffice_4.3.4_Linux_x86-64_deb.tar.gz

Go add the re-add the applications to your launcher!

Some Kubuntu users will need to run this command to fix how LibreOffice 4 looks in KDE:

sudo mv /opt/libreoffice/ure/lib/libstdc++.so.6 /opt/libreoffice/ure/lib/libstdc++.so.6.old  

If you are running a newer version of Ubuntu you can get various versions of LibreOffice 4 right from the repos:

• 14.10 has LibreOffice 4.3.3

• 14.04 has LibreOffice 4.2

• 13.10 has LibreOffice 4.1

• 13.04 has LibreOffice 4.0.2

Install it with:

sudo apt-get install libreoffice  

Or use the software center.

If you want the absolute latest version of LibreOffice follow this answer.

You can install the beta packaged for Ubuntu from the LibreOffice pre-releases ppa:

https://launchpad.net/~libreoffice/+archive/libreoffice-prereleases

Via terminal:

sudo add-apt-repository ppa:libreoffice/ppa -y
sudo apt-get update
sudo apt-get install libreoffice -y

By using the official ppa you'll always have the latest LibreOffice stable release, regardless your Ubuntu version (at least until it become obsolete, so 14.04, 12.04 and the upcoming 15.04 are fully supported).

Instructions are on this Web Upd8 article: LibreOffice 4.0 Available For Download.

Differs from Seth's answer only in copying the desktop integration .deb from one location to another before the dpkg command.