The feasible set in a conic program is the intersection of a convex cone with an affine space. In this talk, I will be interested in the feasibility problem of conic programming: How to decide whether an affine space intersects a convex cone or, conversely, that the intersection is empty? Can we compute certificates of infeasibility? The problem is harder than expected since in (non-linear) conic programming, several types of infeasibility might arise. In a joint work with R. Sinn we revisit the classical facial reduction algorithm from the point of view of projective geometry. This leads us to a homogenization strategy for the general conic feasibility problem. For semidefinite programs, this yields infeasibility certificates that can be checked in polynomial time. We also propose a refined type of infeasibility, which we call "stable infeasibility” for which rational infeasibility certificates exist.