These notes have grown out of a set of lectures given at the Universite de Liege in Spring 1983 for the 3eme Cycle FNRS en Physique Nucleaire. When several members of the theory group at Liege suggested that it would be nice to have the lecture notes available in printed form, I thought this would be an easy thing to do. As the writing-up progressed, however, it became increasingly clear that even for some of the basic questions about the physics of the quark-gluon plasma no definite answer was available in the literature. Some of these questions are: Of what type is the phase transition from hadronic to quark matter? Can there be overheating or undercooling resulting in large fluctuations? How does a fireball consisting of quark-gluon plasma finally break up: by bulk expansion, by surface radiation, or by successive fissioning? What is the best experimental signature for the existence of the quark-gluon plasma, with quantitative predictions including background processes?