How do Tumblr theme updates work?

There are two paths to updating your theme. The first, and easiest, is if you haven't customized your theme code. If the theme code has been left untouched then your theme will automatically update without prompting you.

If you have made changes to your theme's HTML, you can reinstall your theme by locating it from the  Browse themes link in your customization options. Locate and select your theme, and finally click the Use button in the top right corner of the sidebar. This will grab the most recent version of the theme and overwrite any changes you may have made.