This discussion is basically the entirety of Bava Kama. There's way too much to source directly, but I'll try to point to a couple things.
The source for Chovel (recently came up in a Daf Yomi Shiur for Kesuvos 32) is derived from the possuk that ossurs a shaliach beis din from adding to the malkus. Chazal point out that Beis Din is given the authority to wound only so far, and that if a shaliach commits an aveirah even by going over the permitted number of malkot, then everyone NOT approved by Beis Din is certainly forbidden from intentionally injuring another person.
BONUS: As mentioned in 1, Torah law does not recognize a voluntary waiver against personal injury. This means that boxing a fellow Jew is absolutely forbidden. (Fun Rashi on parshas shemos 2:14 - Dathan and Aviram were called risha'im merely for lifting their hands to strike! Sanhedrin 58b) However, their is a question about whether wounding a non-Jew is prosecutable to the same extent as a Jew. The possibility exists that such a waiver, for a non-Jew, might be binding, making it permitted to hit him. He would still be held liable for hitting you, however, since your corresponding waiver would be invalid. Mipnei darchei shalom, we tend to treat non-Jews and Jews in the public domin identically.
As @user6591 hinted at, there IS an exception written into halacha regarding accidental personal injury in specific circumstances where we expect the individual to be "out of his mind" - injury by a drunkard on Purim is the big one here. But that doesn't provide an exemption for proactively entering into a fight with a clear mind and the express intent to injure another person.
Chazal would likely not condone the Jewish boxer/MMA fighter, especially since it's a dangerous sport where you subject yourself directly to injury - a violation of Vinishmartem Me'od Linafshoseichem. This piece has a decent treatment of it.
To answer @msh210's objection: The Rambam's psak regarding providing a waiver is quoting the psak in the mishna at the end of the 8th perek in Bava Kama. Rabbi Akiva in a previous mishna in that same perek (where the psak is like him) brings this point up to prove that a "Chovel be'atzmo" is still chayiv for damaging himself, even though there is no payment of any of the five forms of nezek. Kal VeChomer someone damaging another "with permission" is still considered assur.