I'd like to add my 2 cents worth as a someone who is training in HEMA -Historic European Martial Arts
Firstly, I must dispel the myth of the heavy, blunt, crude european sword. European swords were as finely forged as any Japanese sword, and were often made with better steel. Take the longsword for instance. A longsword could be up 4 feet long, but weighed only slightly more than a Katana. The European one handed Arming sword, what people mistakenly call a 'broadsword' was often significantly lighter than a Katana.
Like all swords (including the Katana) they were sharpened to the level of a wood chisel, although they could, and sometimes were, honed to a razor sharpness.
Secondly, European swords were designed to cut flesh, not armour. Plate armour is not only very light (weighing in at about 65 pounds), it is also nearly impossible to cut with a sword. If you wanted to smash away at armour you used a polearm, a mace or a warhammer.
There were two main ways to use a longsword. The first way is blossfechten. Blossfechten is Bare Fighting, and was used against lightly armoured foes. This style is similar to everyones classic notions of how swordfighting is, with cuts, parries etc etc. Blossfechten was swift and deadly, with most fights over in a few exchanges.
YouTube - Zornhau training - Lichtenauers longsword techniques
The second way is Harnischfechten, Harness Fighting. Since plate is almost impossible to cut, you must pierce the weakspots. In order to do this the blade was grasped halfway down in length in a technique known as halfswording. The extra leverage and accuracy allowed the blade to stab into the seams of the armour. In addition the blade could be reversed and and the extending quillions of the hilt used like a mace to smash the face and the helmet. Harnischfecten was closely integrated with grappling, and fights often ended with daggers.
YouTube - SwArta Harnischfechten
There were also systems of battlegrappling known under the umbrella term Kampfringen, which translates to battle grappling. Grappling was closely integrated into weaponised combat.
YouTube - Hip Throw - Ringen am Schwert ("Wrestling at the Sword")
YouTube - ringschule Wroclaw
YouTube - Twirch Ringen (Updated)
We know a great deal about the fighting styles of the late medieval era because many of the fighting men wrote down what they knew in Fechtbuchs "FightBooks". These describe fighting techniques for everything from swords, to sickles, to clubs, to dueling shields, armed and unarmed. There are dozens known, with a concerted search underway to find more. The best known are Fiore Dei Liberi's Flos Duelletorum the Tower Manscript, which describes sword and buckler, the works of Talhoffer, Joachim Meyer, Liechtenauer and Paulus Hector Mair.
Since Mairs is the largest and finest illustrated, I'll link to it. The rest can be easily found with some googlefu.
Digitale Bibliothek - M