Edit: Due to the fact that this article is old, the installation instructions may not work anymore. In order to successfully install the latest version of LibreOffice, please access the libreoffice tag and open the latest article (the one on top).
The latest installation instructions for LibreOffice
Hello Linux Geeksters. LibreOffice 4.1.2 has been added to the LibreOffice PPA, and is available for Ubuntu 13.04 Raring Ringtail, Ubuntu 12.04 Precise Pangolin and derivates. 
In this article I will show you how to install LibreOffice 4.1.2 on Ubuntu 13.10 Saucy Salamander, Ubuntu 13.04 Raring Ringtail, Ubuntu 12.10 Quantal Quetzal, Ubuntu 12.04 Precise Pangolin, Linux Mint 16 Petra, Linux Mint 15 Olivia, Linux Mint 14 Nadia, Linux Mint 13 Maya, Debian Jessie, Debian Sid and Debian Wheezy, Pear OS 8, Pear OS 7, Elementary OS 0.2 Luna and other Debian and Ubuntu derivates.
For Ubuntu 13.04, Ubuntu 12.04, Elementary OS 0.2 Linux Mint 15, Linux Mint 13 and Pear OS 8, LibreOffice 4.1.2 is available via PPA, so installing it is easy. Just do:
$ sudo add-apt-repository ppa:libreoffice/ppa
$ sudo apt-get update
$ sudo apt-get install libreoffice
To install LibreOffice 4.1.2 on Ubuntu 13.10, 12.10, Linux Mint 16, 14, Pear OS 7, Debian Sid, Debian Jessie and Debian Wheezy, we have to download the archive from the official site and install the deb packages contained by the archive, like we did here. Follow the instructions for your system’s architecture exactly, in order to get a successful installation:
How to install LibreOffice 4.1.2 on 32 bit Ubuntu, Linux Mint, Debian and Pear OS:
$ wget -c download.documentfoundation.org/libreoffice/stable/4.1.2/deb/x86/LibreOffice_4.1.2_Linux_x86_deb.tar.gz
$ tar -xzvf LibreOffice_4.1.2_Linux_x86_deb.tar.gz
$ cd LibreOffice_4.1.2.3_Linux_x86_deb/DEBS
$ sudo dpkg -i *.deb
How to install LibreOffice 4.1.2 on 64 bit Ubuntu, Linux Mint, Debian and Pear OS:
$ wget -c download.documentfoundation.org/libreoffice/stable/4.1.2/deb/x86_64/LibreOffice_4.1.2_Linux_x86-64_deb.tar.gz
$ tar -xzvf LibreOffice_4.1.2_Linux_x86-64_deb.tar.gz
$ cd LibreOffice_4.1.2.3_Linux_x86-64_deb/DEBS
$ sudo dpkg -i *.deb