We develop a version of homological algebra for semimodules which enables us to:
(1) introduce new cohomology monoids of an arbitrary monoid M with coefficients in semimodules over M as more computable alternatives to the old ones,
(2) construct singular homology and cohomology monoids of topological spaces with coefficients in abelian monoids so that the homotopy axiom holds,
(3) generalize the construction of derived functors via simplicial resolutions to semimodule-valued functors. Some other applications of our approach are also presented.