Efficient algorithmic computation of homology groups and the fundamental group of a subset of Euclidean space is among the basic problems in present-day Applied Topology. Particular interest is in algorithms taking finite CW complexes on input. At the end of 20th century Robin Forman proposed a version of the classical Morse theory for a CW complex embedded with a combinatorial counterpart of a vector field. In the talk we will demonstrate how this theory may be fruitfully used in the construction of homology algorithms. We will also show a recent extension of this approach to the algorithmic computation of a presentation of the fundamental group.