When dealing with the Japanese katana I know of three.
This is basically Japanese fencing, a sport. The weapon you would use in this style is a shinai. The techniques used are derived from Kenjitsu.
This is a practical combative sword art in where techniques are learned against a variety of different weapons. Generally you will practice with a partner in sparring forms or sets utilizing bokuto (bokken) as a substitute for a real sword.
This is a sword art that deals primarily with the proper drawing of the sword, cutting techniques, removing the blood from the blade and the re-sheathing of the sword. It is a very smooth and precise discipline. You would likely start by using a bokuto, as you become more advanced an unsharpened sword called an Iaido would be used and finally when you become advanced you would use a real katana.
This short list and its descriptions are by no means comprehensive and I'm not an expert I've just done some research on the topic and this is what I know.
Besides fencing, I know of no other arts dedicated solely to a specific type of sword. When it comes to other types of swords (since you didn't specify what type of sword you want to learn to use) their use is usually preserved in various martial arts. For example many styles of Kung Fu will have in their curriculum the broad sword, the straight sword, twin hook swords, etc... However I know of no style that exclusively teaches any of these swords.
I hope this helps :)