This topology course deals with singular homology and cohomology of topological spaces. Homology groups H_n(X), for n = 0,1,2... are abelian groups and they are assigned to a space in a functorial way, i.e. for any continuous map f: X --> Y there are homomorphisms f_*: H_n(X) --> H_n(Y) for n=0,1,2.... Homology groups are in general easier to calculate than homotopy groups, because they have several structural properties (homotopy invariance, long exact sequences for pairs of spaces, additivity, excision etc). Cellular homology, the Mayer-Vietoris sequence and the Künneth-theorem allow many concrete calculations. On the level of cohomology we have the cup-product. This multiplicative structure together with the cap-product that combines cohomology and homology, is a further feature that allows us to use algrebraic means in order to get geometric statements. We will discuss several examples and some geometric applications such as Poincare duality.