The detection of quasars at $z>6$ unveils the presence of supermassive black holes (BHs) of a few billion solar masses. The rapid formation process of these extreme objects remains a fascinating and open issue. Such discovery implies that seed black holes must have formed early on, and grown via either rapid accretion or BH/galaxy mergers. In this theoretical review, we discuss in detail various BH seed formation mechanisms and the physical processes at play during their assembly. We discuss the three most popular BH formation scenarios, involving the (i) core-collapse of massive stars, (ii) dynamical evolution of dense nuclear star clusters, (iii) collapse of a protogalactic metal free gas cloud. This article aims at giving a broad introduction and an overview of the most advanced research in the field.