Your question is way too broad to answer thoroughly. By far the most common type of issue is a theme/plugin that is incompatible with the new WP version. So you want to check that first.
The upgrade from 4.5 to 4.6, for instance, brought more thorough error catching, prompting plugins to stumble over errors that previously went unnoticed.
After checking the compatibility status of your plugins it’s just a matter of backing up your database and pushing the ‘update’ button.
UPDATE Depending on how critical your site is, you could also clone it and update/debug the cloned install first, before moving on to the live server.