The current version which is 2.55.0 is not working for me and does not support my Pocketbook Inkpad 3 ebook reader and other ways of installation are too complicated for me.

Because of the fact that we can not change release policy inside Ubuntu, you probably will not get it from official Ubuntu xenial repositories.

But you can use one of the following methods to install the software:

1. 
   

   You can try to install calibre from Jonathon F's PPA.

   
   
   

   sudo add-apt-repository ppa:jonathonf/calibre
   sudo apt-get update
   sudo apt-get install calibre
   

   
   
   

   This will install v3.25 which is newer than v3.21 from Ubuntu Bionic Beaver 18.04 LTS.

   
   
   

   It looks great:

   
   
   

   

   
   


2. 
   

   Also can try to download latest version from official site:

   
   
   

   cd ~/Downloads
   https://download.calibre-ebook.com/3.26.1/calibre-3.26.1-x86_64.txz
   tar -xf calibre-3.26.1-x86_64.txz
   

   
   
   

   and launch it from here with

   
   
   

   ./calibre
   

   
   


3. 
   

   Other possibility is a Snap package.

   
   
   

   Currently no calibre snap-package exist. Calibre has very beautiful interface, but Snap currently ignores theming, so it will not look pretty.

   
   
   

   And I have doubts about the functionality of this option. Snap-version may not have access to USB-memory of the ebook-reader because of missed removable-media plug/interface.

   
   
   

   Also about Snap version I can cite Eli Schwartz (eschwartz) from "
   feature request - package calibre as snap app" bug-report 1707517 from 2017-07-30 at launchpad.net:

   
   
   

   
   

   There is already a cross-distro tarball that works without snap.

   
   
   

   Also IMHO snap is still a disaster and I'm very grateful Kovid doesn't
   use it. He's already rejected snap multiple times

   
   
   

   https://calibre-ebook.com/download_linux

   
   
   

   duplicate 1699620

   
   

   
   

The answer is "never" (Or One Day...)

Ubuntu 16.04 was released 2 years ago. The software center keeps apps that are tested to be stable and works well with that version.

The software center of a more recent version of Ubuntu, such as 18.04, which came out last week, have a more recent version of Calibre 3.21.0.

You may want to upgrade your Ubuntu to 18.04 in about 6 or 7 month's time when the upgrade option will appear in your Ubuntu 16.04 system. At the time of the upgrade, your Calebre will be updated to version 3.21. It will remain at that version until you upgrade Ubuntu again.

Or One Day...

The makers of Calibre (or someone else) may decide to make their software available as a snap package to Ubuntu. In that case, version 3 or newer may be available to Ubuntu 16.04. The nature of snap package is such that it will get automatically updated to the newest version irrespective of the version of Ubuntu.

Use a PPA until then

A Personal Package Archive (PPA) is a software repository for uploading source packages to be built and published as an Advanced Packaging Tool (APT) repository by Launchpad. As the name implies, these are personal, and not officially supported by Ubuntu. See Are PPAs safe to add to my system and what are some "red flags" to watch out for?

If you want to use the PPA and get Clibre 3.21 running in Ubuntu 16.04, then follow N0rbert's answer and open a terminal by pressing Ctrl+Alt+T and enter:

sudo add-apt-repository ppa:jonathonf/calibre
sudo apt-get update
sudo apt-get install calibre

Or if Calibre is already installed replace the last command with:

sudo apt-get upgrade

After the first command, you will be prompted for your password. When you type the password in the terminal, nothing will show, and the cursor will not move. This is normal. Just type the password blindly and hit enter. In the subsequent times you won't be asked for the password.

Hope this helps