A Rosenthal compactum is a pointwise compact set of functions of the first Baire class on a Polish space. In a sense, the role of Rosenthal compacta inside the larger class of general compacta is similar to the role of analytic sets inside the class of general subsets of Polish spaces. Many pathologies are discarded by their definable nature and structural theorems appear instead. We will review some facts and we will present more recent results in collaboration with S. Todorcevic and with A. Poveda and S. Todorcevic.