What is the latest drush version and how do I install it? I have drush installed using what in principle is the latest stable build as per the instructions. When I run drush version then i get the reply 8.0.3, however on my workstation with Dev Desktop installed it tells me it's 8.0.5.

Running drush dl drush proposes a version 8.x-6.x-dev dated 30 September 2013 which seems very old, and I can see no sign of a v8.1

Looking at the documentation I have not been able to see any other indications of how to go about this, and I am confused. Can anyone tell me whether the drush dl drush is still valid and what the current latest version actually is?
drush dl was never supposed to be used to download Drush. It happens to work because Drush is hosted on drupal.org as module, but the latest development are done on github.com. On drupal.org, you find just old versions, dated 2013.

As per July 17, 2016, the latest version is 8.1.2. You can install it using Composer, if you are using any Linux OS, or OS X, and in this case it is easy as running the following command:

composer global require drush/drush:~8


Don't forget to add ~/.composer/vendor/bin to your shell PATH variable, or the shell will not find Drush's executable.