Abstract: I shall define the Fredholm index (resp. Euler characteristic) of an elliptic PDE (res. elliptic complex) acting on smooth sections of bundles over a closed manifold. This is the analytic index map. I shall also sketch a proof of the fact, due to Atiyah and Jaenich, that the space of Fredholm operators is a classifying space for K-theory. Time permitting, I shall indicate the classifying space of odd K-theory (space of self adjoint Fredholm operators and spectral flow). Basic tools of functional analysis like compact operators and pseudodifferential calculus will be recalled.