Contramodules are module-like algebraic structures endowed with infinite
summation (or, occasionally, integration) operations satisfying natural axioms.
Introduced originally by Eilenberg and Moore in 1965 in the case of coalgebras
over commutative rings, contramodules experience a small renaissance now after
being all but forgotten for three decades between 1970-2000. Here we present a
review of various definitions and results related to contramodules (drawing
mostly from our monographs and preprints arXiv:0708.3398, arXiv:0905.2621,
arXiv:1202.2697, arXiv:1209.2995, arXiv:1512.08119, arXiv:1710.02230,
arXiv:1705.04960, arXiv:1808.00937) - including contramodules over corings,
topological associative rings, topological Lie algebras and topological groups,
semicontramodules over semialgebras, and a "contra version" of the
Bernstein-Gelfand-Gelfand category O. Several underived manifestations of the
comodule-contramodule correspondence phenomenon are discussed.Comment: LaTeX 2e with pb-diagram and xy-pic; 93 pages, 3 commutative
diagrams; v.4: updated to account for the development of the theory over the
four years since Spring 2015: introduction updated, references added, Remark
2.2 inserted, Section 3.3 rewritten, Sections 3.7-3.8 adde
