  The review is devoted to the elucidation of the basic problems arising in the
theoretical investigation of systems with Bose-Einstein condensate.
Understanding these challenging problems is necessary for the correct
description of Bose-condensed systems. The principal problems considered in the
review are as follows: (i) What is the relation between Bose-Einstein
condensation and global gauge symmetry breaking? (ii) How to resolve the
Hohenberg-Martin dilemma of conserving versus gapless theories? (iii) How to
describe Bose-condensed systems in strong spatially random potentials? (iv)
Whether thermodynamically anomalous fluctuations in Bose systems are
admissible? (v) How to create nonground-state condensates? Detailed answers to
these questions are given in the review. As examples of nonequilibrium
condensates, three cases are described: coherent modes, turbulent superfluids,
and heterophase fluids.
