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The simplicial volume is a homotopy invariant of manifolds introduced byGromov in 1982. In order to study its main properties, Gromov himself initiatedthe dual theory of bounded cohomology, that developed into an active andindependent research field. Gromov's theory of bounded cohomology was based onthe use of multicomplexes, which are simplicial structures that generalizesimplicial complexes without allowing all the degeneracies appearing insimplicial sets.  In the first part of this paper we lay the foundation of the theory ofmulticomplexes. We construct the singular multicomplex K(X) associated to atopological space X, and we prove that K(X) is homotopy equivalent to $X$ forevery CW complex X. Following Gromov, we introduce the notion of completeness,which translates into the context of multicomplexes the Kan condition forsimplicial sets. We then develop the homotopy theory of completemulticomplexes.  In the second part we apply the theory of multicomplexes to the study of thebounded cohomology of topological spaces. We provide complete proofs ofGromov's Mapping Theorem (which implies that the bounded cohomology of a spaceonly depends on its fundamental group) and of Gromov's Vanishing Theorem, whichensures the vanishing of the simplicial volume of closed manifolds admitting anamenable cover of small multiplicity.  The third part is devoted to the study of locally finite chains onnon-compact spaces. We expand some ideas of Gromov to provide complete proofsof a criterion for the vanishing and a criterion for the finiteness of thesimplicial volume of open manifolds. As a by-product of these results, we provea criterion for the l^1-invisibility of closed manifolds in terms of amenablecovers. As an application, we give the first complete proof of the vanishing ofthe simplicial volume of the product of three open manifolds.

Gromov's theory of multicomplexes with applications to bounded cohomology and simplicial volume