Spin geometry arises from the attempt to define a first-order differential operator whose square is equal to the Laplace operator. In Euclidean space this problem can be solved after moving from scalar valued functions to functions taking values in Clifford representations. This construction is carried over to Riemannian manifolds and leads to the definition of Dirac operators. We will explain the main steps of this construction, study analytic properties of Dirac operators, and outline the famous Atiyah-Singer index theorem. These lectures provide some background for the lectures of Christian Bär, who will study the implications in scalar curvature geometry.