According to the Nash-Tognoli theorem, each compact smooth manifold M is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of M. It is interesting to investigate to what extent algebraic and differential topology of compact smooth manifolds can be transferred into the algebraic-geometric setting. Many results, examples and counterexamples depend on the detailed study of the homology classes represented by algebraic subsets of X, as X runs through the class of all algebraic models of M. The present paper contains several new results concerning such algebraic homology classes. In particular, a complete solution in codimension 2 and strong results in codimensions 3 and 4.