At high temperatures, strongly interacting matter is expected to undergo a phase transition to a "quark-gluon plasma" (QGP), a state of matter that is believed to have existed in the first few microseconds after the Big Bang. By colliding heavy ions at ultrarelativistic velocities, physicists can create and study the properties of the QGP and high-temperature hadronic matter. I will give an introduction to the unique and unexpected phenomena we observe in these collisions, including hadron suppression, jet quenching, and hydrodynamic behavior. I will show how hadronic resonances and heavy quarkonia can be used be used to probe these systems and study their properties, moving us towards a better understanding of the nuclear equation of state. Finally, I will offer a glimpse of what lies ahead for this field at RHIC and the LHC, as well as at the next generation of colliders.