There are several sports/groups/ways to experience sword fighting in modern day life. The most well known of these are the sports of fencing and kendo. They are sport versions of rapier and katana fighting. On the whole they are the most unrealistic ways to experience sword fighting. They use set strikes and have only a few legal targets. They are so formalized they are much more sport then combat, IMHO.
Another way to experience sword fighting is through the use of boffer weapons. These are used mostly by LARPs and you can find information of making these on several sites devoted to boffer LARPs such as nero and solar. These weapons are rather light and way too flexible, but they are cheap and easy to make. Also you don't need to use any sort of padding or armor. And they aren't any where near as intimidating as say, SCA weapons, so it's easier to find people to play with.
The SCA is a medieval recreationist group that also does heavy fighting. This is fully armored combat, with participants making (or buying) their own armor. The weapons as are made from rattan, a relative of bamboo. Combat can quite often leave bruises especially if you're not wearing much armor. Most fighters in the SCA use sword and shield although they there are some people who use other styles such as two handed sword, florentine (two weapons), mace, axe, and pike / polearm. One of the best points about SCA fighting is it is most likely the largest melee fighting you can find, with hundreds of people in some battles.
The Historical Armed Combat Association is a lose group of people who wish to study European arms and fighting. It encourages several different types of sparring and practice fighting - from full contact with padded swords to edge testing with a live sword (not on people). They strive to be as historically accurate as possible, while still being safe.
The Academy of European Medieval Martial Arts (or A.E.M.M.A.) is a group dedicated to recreated European martial arts and period training methods. They do several kata like drills called folgen, then advancing to planned and free form sparring. They also train some unarmed and non-sword weapons. They have a few different ranks in both weaponsmanship and rank of the student as a whole. They are currently working on a book detailing swordsmanship and their approach to training.