Tournaments are traditionally considered to be invented in 1066.

Now, duelling and jousting are two very different things. A duel is essentially a combat between a social/legal offender and the one offended. Jousting is a martial art carried into a sport form, of riding hard at each other with lances in order to unseat the opponent. Duels are normally fought with other weapons, as lances in a joust are rarely decisive.

Tournaments began with emphasizing the melee or mock battle, which might begin with lances, tipped with coronels rather than spearheads to keep down injuries. Plenty still happened, including deaths.

By the time of King René's Book of the Tournament in 1460, he describes a melee with little jousting, fought in cuirboilli armour with whalebone swords, and lots of fancy decorations.

The joust really developed from the 1300s forward, slowly replacing the melee. It had the advantages of requiring less skills and being able to have more safety equipment. Still, Henri II of France was killed in 1559.

Edit: Safety equipment? Definitely. The great helm was held around the head by a system of linen bands like the suspension in a hardhat. A blow on it was not transmitted to the skull. Because blows came only from one place, they could lighten back armour while making the front and left super heavy (jousting harness could be twice the weight of battle harness). All this from ffoulkes, which you can pick up at Project Gutenberg or Internet Archive.

In the appendices of The Armouer and His Craft, he includes a description of a judicial duel of the Middle Ages. So, yes, they were fought.