Brian Switek, writing over at Nature, has a fantastic summary of what scientific questions Tyrannosaurus researchers are interested in, focusing in on four big ?s.
1) How did Tyrannosauroidea become Tyrannosauridae?
The earliest members of the theropod group containing Tyrannosaurus were a few meters in length (e.g. Dilong, illustrated above) and were not the dominant predators of any food chain. But Tyrannosauroidea stayed in place while other theropod lineages died out, eventually allowing Tyrannosauridae to include the largest theropods of the late Cretaceous in Asia and North America. What allowed them to do so? Good genes? Lucky timing?
2) What did Tyrannosaurus look like when young?
Nanotyrannus (pictured above) may or may not be "Nanotyrannus": it might be a juvenile Tyrannosaurus, or it might be a distinct genus. The life of Tyrannosaurus from hatching to adulthood is not as clear as we wish it was, so it's not completely certain when, or how, an individual Tyrannosaurus transitioned from a fairly-sleek theropod into the massive adult morphology that we all know and admire (sometimes from afar).
3) Did Tyrannosaurus have feathers, fluff, or neither?
Early members of Tyrannosauroidea have flight-incapable feathers (similar to a modern bird's down) that would have provided some form of thermal insulation. Tyrannosaurus, as an adult, weighed as much as an elephant and lived in a climate warmer than any elephant: it did not need thermal insulation. Did all members of Tyrannosauridae lack feathers/fuzz/fluff? Was it lost during an individual's life or was Tyrannosaurus always sans plumage?
4) What's the deal with the arms?
They're small in comparison to the adult body size. But they're not lacking in musculature: if Tyrannosaurus lifted weights, it would probably lift more than you could. Maybe the arms were used for social signals more than for any other activities, it's tough to tell and this might be a question not able to be answered.
All in all, I think Mr. Switek, who usually writes here, did a great job, and I recommend reading his review, of which this is a mere appetizer.