Your question is a common one these days. In the 1990s, paleontologist Jack Horner argued that T. rex was a scavenger, only eating animals that had already died, rather than actively hunting and killing them first. Because this claim contradicted the common notion that T. rex was a vicious predator, the media—from newspapers to radio to television—quickly picked up on the story, leading many people to think that science favored the "T. rex was a scavenger" theory.
However, of the paleontologists that I have met, it seems that only Jack subscribes to the theory. I, too, disagree with Horner; the arguments that I've seen him put forward for a scavenging T. rex are not strong. For example, he states that the short arms of T. rex were useless for hunting. True, but don't forget about the massive skull that's armed with a mouth full of teeth. The impact and bite from such jaws were no doubt quite effective in taking down prey, so who needs arms? Another argument that Horner commonly presents is that T. rex was not a fast runner, so it couldn't have successfully chased and captured its prey. Indeed, T. rex was probably no speed demon (see my answer to one of the previous questions), but neither were its potential prey items, the "duck-billed" hadrosaurs and horned ceratopsians, so T. rex did not need to be a fast runner—only fast enough.