When Yitzchak Avinu prepares to give his Beracha to Eisav, he instructs him to go out to the field and hunt some game for him, and then to make it into a meal for him to eat (בראשית כ"ז:ג'-ד'). This request would appear to be consistent with Yitzchak's already recorded tendency to favor Eisav (over Yaakov) because of Eisav's ability to hunt and trap game which Yitzchak would then eat (שם כ"ה:כ"ח), an ability which, according to the previous Posuk (שם פסוק כ"ז), indeed identified Eisav's very personality. Although Rashi (שם בד"ה יודע) quotes a view from the Midrash Tanchuma on our Parsha (אות ח') that interprets this ability figuratively, meaning that Eisav had the ability to "trap" people, that is, to ensnare or trick them with his words and thereby deceptively represent himself as something other than what he really was, it is clear from Rashi's next comment (שם בד"ה איש) that Eisav was in fact a hunter of game, whether animals or birds, in the literal sense as well, and hence Yitzchak's aforementioned request of him prior to giving him his Beracha. It is noteworthy that earlier in the Torah (שם י':ט'), Nimrod is similarly described as a mighty hunter; there too, Rashi (שם בד"ה גבור), quoting from the Midrash in Bereishis Rabbah (פרשה ל"ז סימן ג'), offers the figurative interpretation, explaining that Nimrod ensnared people with his words and incited them to rebel against Hashem. The Midrash itself (שם) actually draws the connection to Eisav in this regard. It is clear from the Midrash Aggadah (לפסוק ח' שם), however,which indicates that until the days of Nimrod nobody ate meat, that Nimrod was indeed a hunter of animals in the literal sense as well.
It is evident from the Midrash Aggadah (שם) that one of the purposes of hunting is to be able to have meat to eat; it was obviously the intent of Yitzchak, when he asked Eisav to go hunting, that Eisav should be able thereby to prepare him something tasty to eat, as the Pesukim state clearly (שם כ"ז:ד'-ה'). The Torah much later (ויקרא י"ז:י"ג), in a totally different context, also speaks of hunting, or trapping, an animal or a bird for the purpose of eating the meat. It is clear, however, from one of the comments of Rashi on our Parsha cited above (לבראשית כ"ה:כ"ז בד"ה איש), that hunting, if done not for food, but for the amusement or the sport of it, as explained by Rabbeinu Eliyahu Mizrachi (שם), is a worthless activity, pursued only by people who have nothing worthwhile with which to occupy their time. This seems to be the view expressed as well in the Targum Onkelos on that Posuk (שם), as understood by Tosafos in Bava Kamma (דף צ"ב: בד"ה דשרכי) and in Bava Basra (דף קל"ט. בד"ה מתני'). Hunting, therefore, solely for the purpose of pleasure or entertainment, seems to be viewed by Chazal as a valueless activity and a waste of time.
This description, of course, does not necessarily mean that hunting is forbidden by Halacha, but there are certain sources which indicate that hunting for sport is an activity that should indeed be avoided. The Midrash in VaYikra Rabbah (פרשה י"ג סימן ג'), as explained by the Eitz Yosef (שם בד"ה קניגין), states that in the future, there will be some sort of an opportunity for the Tzaddikim to hunt the Leviassan, the giant fish out of whose meat a great feast will be prepared as a reward for the Tzaddikim in Olam HaBo, but that this opportunity will be made available only to those who do not participate in hunting for pleasure in this world. This would imply that hunting for sport is an activity that may not be pursued, as only those who avoid it are rewarded in this way in Olam HaBo; the Eitz Yosef (שם בד"ה וכל), adds that such activity leads to Bittul Torah and is a waste of time. Similarly, the Gemara in Avodah Zarah (דף י"ח:), according to the interpretation of Rashi (שם בד"ה קיניגון), considers hunting for pleasure to be a sinful activity, alluded to in a Posuk in Tehillim (א':א'). The Rambam, in his Moreh Nevuchim (חלק ג' פרק י"ז), states clearly that it is forbidden to kill animals for the purpose of practicing cruelty or for the sport of it, while the Sefer HaChinuch (מצוה קפ"ו), in discussing the prohibition to slaughter animals dedicated for Korbanos outside of the Beis HaMikdash, writes that it is prohibited to kill an animal for no purpose and that the wanton slaughtering of animals is similar in some ways to murder.
The Maharsha, commenting on the aforementioned Gemara in Avodah Zarah (בחדושי אגדות שם בד"ה לטרטיאות), categorizes hunting as simply a silly and frivolous activity; the Ramo, commenting on the Tur in his Darkei Moshe (אורח חיים סימן שט"ז אות ב'), writes that it is prohibited to go hunting with hunting dogs because this is a foolish activity, and one who does so loses this reward relating to the Leviassan in Olam HaBo. The Ramo thus rules in the Shulchan Aruch (או"ח שם סעיף ב') that it is forbidden to hunt with hunting dogs because of the frivolous nature of the activity. The Pri Megadim (באשל אברהם שם ס"ק ה'), though, notes that this is the case only if one does this simply to occupy his time, but not, perhaps, if he makes his living that way; the Aruch HaShulchan (שם סעיף ב') similarly outlaws hunting only if it is done with hunting dogs, but rules that it is permissible for one to hunt by himself if it's for his own needs, such as for financial gain or for food. The Darkei Teshuvah (יורה דעה סימן קי"ז ס"ק מ"ד), however, rules that people who go hunting with hunting gear, whether for the purpose of making money by selling what they catch or simply for the pleasure and the amusement of it, are violating a prohibition; those who do sell what they catch violate the prohibition against benefitting financially from a forbidden activity, while those who hunt for sport are following the path of Eisav, behaving with cruelty by killing Hashem's creatures for no reason, when they should rather engage in building up the world, not in destroying living things for sport.
In a lengthier discussion about this subject, the Noda BeYehudah (שו"ת נודע ביהודה מהדורא תנינא חלק יו"ד סימן י') first demonstrates that by hunting, one would not violate the prohibition, codified in the Shulchan Aruch by the Ramo (חושן משפט סימן רע"ב סעיף ט') as being MideOraisa, against causing צער בעלי חיים, pain to living creatures, since that prohibition does not apply when the creature is being killed; this is a view which he believes is alluded to by the Gemara in Chulin (דף ז:) that does not outlaw killing certain animals based simply on the prohibition of צער בעלי חיים, and which he reiterates elsewhere (שם סימן י"ג ובמהדורא קמא שם חלק יו"ד סימן פ"ג). Although some authorities, like the Ran in Chulin (דף ה: בדפי הרי"ף בד"ה מתני'), the Bach in his commentary on the Tur (יו"ד סימן קט"ז בד"ה משקין), and Rav Yaakov Emden שו"ת שאילת יעב"ץ חלק א' סימן ק"י(), among others, disagree, this position about םייח ילעב רעצ seems to be accepted by, among others, the Sefer HaEshkol (חלק ג', הל' שחיטת חולין סימן י'), the Yam Shel Shlomo in Bava Kamma (פרק י' סימן ל"ז) who allows poisoning an animal, as cited in the Shulchan Aruch by the Taz (יו"ד שם ס"ק ו'), and, perhaps, the Chasam Sofer (שו"ת חתם סופר חלק יו"ד סימן ק"ג). The Seridei Eish (שו"ת שרידי אש חלק ג' סימן ז'), in discussing the question of using animals for medical experimentation, which he ultimately allows, brings a proof to this position from both a Gemara in Avodah Zarah (דף י"ג:) that suggests that one may kill an animal which has been dedicated to the Beis HaMikdash at a time when there is no Beis HaMikdash, and which therefore can not be used for either its intended purpose or for one's personal benefit, as well as from a Tosafos in Bava Basra (דף כ. בד"ה כיון) which implies that an entire living animal can be thrown to the dogs to be eaten. Hunting, then, would apparently not automatically involve one with the prohibition against causing םייח ילעב רעצ .
The Noda BeYehudah (מהדו"ת שם סימן י') then asserts that by hunting, one also does not violate the prohibition of Bal Tashchis, causing purposeless destruction of things which are useful, as codified in the Shulchan Aruch (יו"ד סימן שמ"ט סעיף ד'), because, among other suggested reasons, one can make use of the dead animal's skin. He does conclude (שם), however, that he finds it difficult to even entertain a question about hunting because hunting was the activity of Nimrod and of Eisav, not of the descendants of Avraham, Yitzchak, and Yaakov; he adds simply that it is improper for a Jew to kill any living creature needlessly or just to occupy his time. He also notes that there may indeed be an actual prohibition involved with hunting because there is great danger in going out "into the wild" to hunt, and one is forbidden to endanger his life, as stated in the Shulchan Aruch (חו"מ סימן תכ"ז סעיפים ט'-י', ועיין שם ביו"ד סימן קט"ז סעיפים א'-ה'); he points out that even Eisav himself considered his life to be in danger when he went hunting, as implied by a Posuk in our Parsha (בראשית שם פסוק י"ב), according to Ibn Ezra's interpretation (שם). The Noda BeYehudah (שם) thus rules that hunting is both frivolous and dangerous and is therefore forbidden. It should be noted, though, that the Ramban in Avodah Zarah (חדושי הרמב"ן לדף י"ג: שם בד"ה והא) writes that one may slaughter animals if it is done for some legitimate human need. Rav Moshe Feinstein (שו"ת אגרות משה חו"מ חלק ב' סימן מ"ז אות א'), however, while ruling that one may kill insects and rodents which are bothersome, concludes that it is preferable to do so indirectly, because killing anything can have a negative impact on one's nature, causing one to develop a cruel personality.
As a postscript, it should be pointed out that one of the 39 prohibited activities on Shabbos is צידה, hunting or trapping, as presented by the Mishnah in Shabbos (דף ע"ג.). The Mishnah later in Shabbos (דף ק"ו:) states that one violates this prohibition if he closes the door to his home and thereby traps an animal inside; the Rambam (פרק י' מהל' שבת הלכה כ"ג) and the Shulchan Aruch (או"ח שם סעיף ה') rule accordingly. The Gemara in Beitzah (דף כ"ד.), however, distinguishes between different types of animals in determining whether the prohibition against צידה applies in a particular case or not; the Shulchan Aruch (שם סעיף י"ב) writes that wild animals or birds which are in one's possession, meaning, as the Mishnah Berurah (שם ס"ק נ"ב) and the Kaf HaChaim (שם אות ק"ט) explain, that they have become domesticated, are used to being in one's home, and are relatively tame, may be "trapped" on Shabbos. The reason for this, as the Mishnah Berurah (שם ס"ק נ"ג) notes, is that these animals are considered "trapped" already by themselves since they return to the home at night and are therefore easily "caught;" the prohibition of צידה thus does not apply, although the Taz (שם ס"ק י"א) points out that one cannot handle these animals because, as the Mishnah Berurah (שם ס"ק נ"ד) explains, all living creatures are Muktzah on Shabbos. The Ramo (שם), however, cites those who disagree and hold that such creatures, if they are out loose, may not be trapped on Shabbos; the Mishnah Berurah in his Biur Halacha (שם בד"ה וי"א) summarizes the different opinions on this matter, concluding that it is preferable to follow the stricter view and avoid trapping such animals on Shabbos, possibly because they are wilder by nature. The Ramo (שם) then implies, though, after saying that those animals which are out of one's control may definitely not be trapped on Shabbos based on Torah law, that those animals which are so inherently tame that they require no trapping at all in order to control, as the Mishnah Berurah (שם ס"ק נ"ט) explains, may be "trapped" on Shabbos provided that they are indeed still tame. One who has a pet in one's home should thus determine how tame and obedient the pet is in order to figure out whether or not he may put a leash on it or close the door with it inside on Shabbos without violating this prohibition.