The heydey of the hunting crossbow in Europe was the 15th through to the 17th centuries. Steel crossbows appeared in Europe in the 14th century, which led to compact, powerful, weather-resistant crossbows, suitable for use on horseback. Also required for use on horseback were suitable spanning devices; the cranequin and goat's foot lever appear (or at least become widely used) in the 15th century. The 17th century saw the development of reliable and accurate flintlock guns that started to displace crossbows for some of their hunting purposes.
Crossbows were used for hunting both before and after that. Earlier hunting crossbows were often cheap wooden weapons, spanned by stirrup and belt-hook. It's with the introduction of crossbows better suited to use on horseback that we see the crossbow becoming a major aristocratic hunting weapon; these were often expensive and decorated weapons, with steel or composite prods. Crossbows continued in use as common hunting weapons into the 19th century.
Some further points:
Stone crossbows, i.e., crossbows made to shoot stones or metal or fired clay bullets, were usually specialised hunting weapons, ideal for small game, against which they would be effective without causing excessive damage.
Crossbows were used in hunting traps. The crossbow could be spanned, and then triggered by the animal. This use probably began quite early, and continued into the 19th century in parts of Europe. This was also done outside Europe, for both small games and large dangerous game (e.g., tigers).
Dogs were trained to retrieve crossbow bolts that had missed their targets.
One sporting use of crossbows was target shooting at "birds" (bird-shaped targets) on poles, mimicking the shooting of birds in trees.
Crossbows were used in Europe for whaling; this continued into the early 20th century.
Much of this (and some further detail on crossbow hunting outside Europe) can be found in Howard L. Blackmore, Hunting Weapons, Barrie and Jenkins, London, 1971, which has a 44 page chapter on crossbows.