Mathematical billiards are an idealisation of the billiard game, played in "tables" of different shapes. They arise naturally in the study of several problems in physics and provide an important model for the mathematical study of chaos. In this talk we will focus in particular focus on polygonal billiards and flows on surfaces, which constitute two important classes of "slowly" chaotic, low complexity systems. We will in particular survey some recent progress on some classical physical systems such as the Ehrenfest billiard (1912) or the Novikov model for electrons in metals.