If you are looking for organized "sports", then kendo and Western fencing are about it. Fencing is a well-established sport that is also an Olympic discipline. The three weapons studied are the foil, the epee, and the saber.
Kendo is the sporting form of kenjutsu, derived from actual sword-fighting techniques and highly formalized. The competitors use extensive armor and the "shinai", a split-bamboo "sword" that's the Japanese equivalent of a foil or epee.
You might look into SCA (Society for Creative Anacronism) fighting... This has been going on for years and seeks to emulate mostly medieval European weapons and techniques using padded weapons.
Also, "LARP" fighting... Live Action Role Playing. This is quite popular as well as you can get into it without all the trappings that are common to the SCA.
There are a very few groups trying to preserve European medieval fighting styles with live weapons.... So far as I know they are rather rare.
I have no idea what maybe around you but my favorite fencing system is arnis (specifically Lastra largo stye arnis). Look around for escrima, kali, silat, arnis for the Filipino and Indonesian styles. Pankration for thethe greek style. Various forms of shaolin kungfu formovie the chinese. And i have found on the web of a few teachers that teach shillelagh fighting, a family system of irish fighters.
Since I have a streak of just asking questions in my head here is what I've been thinking about recently, I'd really like to learn some martial arts sport, but kendo or fencing seem so much cooler than anything with fists and feet, what other sword-sports (I dont even know the word oh holy hell) are there and what do I have to be like to like them or even be able to start training?
A little side note: I live in Croatia and no offense to my people but this country is just bad, all I've seen in my city is a kendo-club so I have to see what kinds of more exotic sports there may be around here