Consider an algebraic curve in 3-space; when projected generically to a plane, it will acquire a number of double points. This number depends only on the degree and the genus of the curve. It can be defined as the degree of the bisecant variety. Computing degrees of k-secant varieties when the curve is replaced by a surface arbitrarily embedded will be the subject of the lecture. One key difference with the curve case is the fact that we have to work with the Hilbert scheme of k points, instead of the k-th symmetric product, and I will spend some time on the construction of the Hilbert scheme.  The main result I will present is the Lehn conjecture, now a theorem, computing all these numbers for all surfaces in terms of their numerical (complex cobordism) invariants.