As a long running company which gave us loads and loads of fun and awesome games over all of these years, it's no surprise that Capcom also has its big share of awesome music as well.
The following games/series have their own pages:
Strider had quite a few good tunes, but "Grandmaster" wipes the floor with the rest of them. Very suitably epic final boss theme.
Rival Schools: United By Fate already had a kick-ass opening theme in its arcade version with "We'll Protect Our School!". But when the game was ported to the PlayStation, it sported an additional animated intro with an even better theme in "Atsuki Kodou."
Resident Evil 5's theme from the chapter 6-3 right before you get to the Bomber Bay to fight Wesker, Majini IX -In Flames, you're running through the burning engine room of an oil tanker sized ship, zombies, zombies with guns, zombies with ROCKET LAUNCHERS all around you, trying to get through two sealed blast doors and a shitton of enemy resistance to get to your goal. The whole damn place is going up in flames and explosions, and THIS is what you rock out to? Yes.
The music for the Crimson Helm bossfight is an adrenaline rush all by itself, and is absolutely perfect for the battle. Likewise, "Waka's Promenade" mixes in the eponymous character's Leitmotif for a song that's twice as awesome. As for different boss music, there's this song, the theme for one of the most memorable Dual Boss battles in all of gaming.
Great Divine Intervention is quite simply one of the most beautiful and inspiring pieces of VGM, ever. The fact that it plays during restorations, the game's best Scenery Porn moments, makes it even mbetter
The fourth game needs a couple nominations at least. "Swipe of Sword" for the obligatory awesome boss theme, and "DRINK IT DOWN" (Wikipedia says it was made for the game), played when you're doing the mandatory install on the PS3, during the OP and in the Bloody Palace randomly. Sadly, only present in the Japanese, and possibly the Asian, version of the game. And here's Swipe of Sword.
While 2 is generally received as a bad game, at least it brought us Heads Or Tails.
The 2009 remake of Bionic Commando, more exactly: the final scene, when Spencer headbutts Super Joe through the cockpit of his mech, makes it explode and falls into the hole - the song is titled "Beyond Dark Skies".
The Last Desperate Struggle gets a special mention, because it stands out among the soundtrack for giving a solid, tense atmosphere, and truly does give a feeling that the finale is just around the corner.