Intersection theory is both the most classical and one of the most rapidly developing area of algebraic geometry. Start with a classical problem: how many lines in three dimensional space meet four lines in a generic position? It turns out that the answer (two lines) can be computed using the Chow ring of a Grassmannian (or can be seen from the following informal argument: degenerate the generic case to a special when we have two pairs of intersecting lines; then we have a line passing through their intersection points and a line that is the intersection of the planes containing these two pairs of lines). Chow rings (or, in the non-smooth case, groups) are an analog of the singular cohomology that are defined over any field, not necessarily algebraically closed or of characteristic 0. The course is devoted to the construction of Chow rings; some examples related to Grassmannians are given.