This book is a direct continuation of the book "Elements of combinatorial and differential topology" .. It begins with the definition of simplicial homologies and cohomologies; numerous examples of their computation and their applications are given. Then KolmogorovAlexander multiplication on cohomologies is discussed. Much of the book is devoted to various applications of (simplicial) homologies and cohomologies. Many of them are related to the theory of obstacles .. One such example is the characteristic classes of vector bundles. Singular homologies and cohomologies are defined in the second half of the book. Then another approach to the construction of cohomology theory is considered - the Czech cohomology and the closely related cohomology of de Rama. The book concludes with various applications of homology theory in the topology of manifolds. In the book there are many tasks (with solutions) and exercises for self-solution .. The book contains a lot of specific material and applications that may be of interest even to specialists in this field. For undergraduate and graduate students of mathematical and physical specialties; for researchers.