Abstract: The Laplacian is arguably the most important among differential operators. Its peculiar properties (e.g. regularity, maximum principles, Fredholmness) actually belong to a wide class of differential operators, which go by the name of “Elliptic operators”. In this talk, we will focus on instances of Dirac operators – elliptic operators that can be thought of as “square roots of a Laplacian” – in geometric contexts. In particular, we will see how, through the Atiyah–Singer Index Theorem, the study of some Dirac operators enables us to prove some main theorems of complex geometry.