Question: I am interested in renaming my characters and worlds. Is that possible? 

I have looked through the UI and could not find a way.

Answer: 
It is possible, but it requires external tools to do so. There are [various inventory editors](http://terraria.gamepedia.com/Inventory_editors) that let you change your player's name and appearance. And, of course, there are tools like [TEdit](http://www.terrariaonline.com/threads/tedit-terraria-map-editor.35394/) that let you edit a world including the name.

Changing the profile/world filename will not rename the world and those files are not in plain text so normal text editor will not do.
I *have* managed to change a worlds name by opening it in a text editor and then just finding and replacing the 2 instances of the old name with the new one. It worked except for that the map display was reset (previously explored areas were cleared) and one other apparent problem was that the new name had to be the same length as the original otherwise the world became corrupt (Terraria failed to load info about the world such as difficulty, size and creation date (showing as "unknown")).

**It is possible to change it without any special tools but I don't recommend it if you don't want to possibly ruin your world.**