The object of the first chapter of this paper is the homological dimension theory of compacta. In this chapter we present Aleksandrov's fundamental results and Bokshtein's theory. A considerable amount of space is devoted to the problem of realization of dimension functions, that is, of constructing compacta with given homological dimensions. In most cases the proofs given here differ from those previously known by certain improvements.In the second chapter we discuss the recently developed homological dimension theory of paracompact spaces. Here the main technical tool is the theory of sheaves. In the last section of this chapter we present Shvedov's theorem, which gives a negative answer to the long-standing problem: is the dimension dim fully determined by Menger's axioms?
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