To begin with, everyone knows that Tyrannosaurus was the largest, most ferocious meat-eating dinosaur ever. If there was one dinosaur you did not want to meet in the primeval jungle, it would be Tyrannosaurus. Everyone else comes in second to its size and danger. Well... Not really, actually. I realize I'll be bursting a lot of bubbles here, but Tyrannosaurus was definitely not the biggest carnivorous dinosaur. Now, don't get me wrong, it was one of the biggest, and certainly was the biggest Tyrannosaur (the rex had many earlier, smaller relatives, such as Daspletosaurus and Tarbosaurus). However, bigger Theropods had come and gone by the time T. rex hit the scene. Dinosaurs like Acrocanthosaurus, Gigonotosaurus, and Charcarodontosaurus were all bigger, unrelated Theropods that prowled the Cretaceous before Tyrannosaurus evolved. These creatures certainly would have been impressive in their time, and all of them would have made Tyrannosaurus look a bit shrimpy. However, the dinosaur Spinosaurus towered over all of them, and was likely the biggest meat eating animal to live on land. Spinosaurus was remarkably longer and taller than Tyrannosaurus, and its height was even further exaggerated by its towering neural spines that formed its characteristic sail. Dino nerds may remember that Jurassic Park 3 (for all its many flaws) did acknowledge that Spinosaurus was bigger than Tyrannosaurus (Dr. Grant in fact claims that it "sounds bigger". I'm not sure how one determines that but oh well). The film goes further to drive this point home by showing the Spinosaurus (spoilers) kill the Tyrannosaurus after an epic battle. While, if this match up were to be hypothetically arranged, seems one sided in that Spinosaurus was indeed the larger of the two, Tyrannosaurus fans can take comfort in the fact that T. rex had a much stronger bit. In fact, it had the strongest bite of any known animal, ever. This is aided by T. rex's long, bone-crushing teeth, contrasting the conical, fish grabbing teeth of Spinosaurus. So the outcome, in more ways than one, would be impossible to determine.
A second misconception involves the seeing abilities of Tyrannosaurus. It's commonly believed that Tyrannosaurus could only see something if it was moving. This is impractical for various reasons, the main one being how an animal who can only see moving objects would be able to avoid trees and rocks and whatnot. But let's assume that it only recognizes prey items by movement. Why on earth would an apex predator who often would have needed to hunt by sight evolve a trait that severely handicapped its ability to find its prey? This makes little evolutionary sense. The truth is that Tyrannosaurus likely had very good eyesight. It had foreword-facing orbits, indicating good binocular vision. It's also reasonable to assume that T. rex had a very good sense of smell, which pokes a hole into the scene in Jurassic Park 1 where the rex is right in front of a very still Dr. Grant and cannot find him. One would assume that, even if T. rex did have terrible eyesight, the sense of smell afforded to most large predators would give away Grants position.
A third rex myth that's popped up, even in the scientific community, is that Tyrannosaurus was not a hunter but a scavenger, feeding off the bodies of dead dinosaurs like a giant vulture. This theory has gained some popularity, but had largely been discredited by most Theropod experts. There doesn't seem to be much of a reason for a scavenging animal to have such a huge size, an extraordinarily powerful bite, teeth evolved for crushing bone in order to kill large animals, and powerful hind limbs. This contrasts the features of most obligate scavengers. Now, many modern large carnivores will scavenge from relatively freshly killed animals when the opportunity arises, and there's no reason to assume Tyrannosaurus was an exception to this rule. However, its anatomical features suggest that it was an animal that was more than capable of bringing down its own prey. Fossilized bones from various large herbivorous dinosaurs have been found with puncture and lesion marks in them. Many of these marks show signs of healing, indicating that the animal was alive when it received the injury and survived long enough for the injury to heal. This, therefore, shows us that Tyrannosaurus did indeed go after live prey.
This, obviously, ruins a lot of people's ideas as to what Tyrannosaurus would have looked like and how it would have behaved. If this is somewhat disappointing, it's important to remember that our view of this dinosaur has changed dramatically over the years since it was first described in the early 1900's. Sadly reality doesn't always concede to how we would like things to be, but at the same time each newly confirmed theory brings us closer to the ultimate truth about how things were and are.