The Noda BiYehuda was famously asked if it is permissible for a Jew to hunt animals for sport. He begins by considering two obvious objections, and rejects them both.
First, he discusses whether hunting would fall under the category of צער בעלי חיים, causing unnecessary pain to the animal. He rejects this objection on two grounds: (1) This rule only applies when there is no benefit for humans, but since when one hunts he stands to get some benefit (perhaps the skins), then there is no consideration of צער בעלי חיים, and (2) this principle only applies when the goal is to apply pain. However, since when one hunts he aims to kill the animal quickly, and to actually put it out of pain, then there is no issue.
The second likely objection is that of בל תשחית, loosely defined as wasting some part of this world. Again, the Noda BiYehuda rejects this problem, as the hunter does stand to gain something once he shoots the animal, and thus there is no wasting involved.
However, the Noda BiYehuda still rejects the permissibility of hunting. He begins the second part of his teshuva by noting that hunting is something that is associated in Tanach with Nimrod and Esav, and thus is not something connected to the character of the descendants of Avraham, Yitzchak, and Yaakov. As to the defense that one should hunt in order to get the skins of the animals, the Noda BiYehuda points out that plenty of animals die on their own and thus there is a readily available supply of such skins without resorting to hunting.
As to the defense that the mishna in Sanhedrin 2a allows for the killing of wild and potentially dangerous animals, the Noda BiYehuda notes that this likely only applies once they have attacked someone and have shown themselves to be actually dangerous, if it applies at all. If anything, hunting for sport is merely revelatory of a cruel nature.
Furthermore, he points out that there may be a specific prohibition involved, namely a violation of ונשמרתם מאד לנפשותיכם - that we are enjoined to take care of ourselves physically. Once who willingly enters a wooded area or some other spot where he is expecting there to be wild and vicious animals is taking his life into his hands (even if he is a relatively good shot), and thus would be violating this mitzva. Even further, the Noda BiYehuda notes that the mishna in the 4th perek of Brachot provides a special prayer that one should recite when entering into a dangerous place, and he reasons that going hunting qualifies as such (based on the fact that even the great hunter Esav said הנה אנכי הולך למות - and the simple explanation is that he expected to eventually die in a hunting accident).