Most quick jokes in French are puns in some way, including the famous "blagues Carambar".

For those who don't know, Carambar is a brand of candy where the wrapper contains a joke. Always regarded a terrible, we love them anyways. Similar to dad jokes in English.

The longer the joke, the less it tends to rely on puns. You don't hear puns very often in stand up comedy for instance.

https://everything2.com/title/Sir+Sufud+and+the+bear

Ages ago, there was a land that was plagued by a terrible bear with a perchance for beating up various small animals that happened to be in the area. This wouldn't be too bad, except that this was in the area with lots of sheep. Now, the sheep themselves weren't in danger, it was the sheep dogs that were threatened.

... (several paragraphs later)

The shepherds, upon hearing the great roar and crash from the cave quickly ran to the place to see a knight sprawled against a tree and a bear underneath it. The dog the bear had been bringing back, however, was alive and well and quite happy to see people again. And so, feeling quite bad for the family of the knight, the shepherds went to the king and offered family of Sir Sufud the dog he had saved from the bear.

Nothing helps the mourning after a terrible knight like the dog of a bear that hit you.