  In this paper, we introduce and investigate \emph{semicorings} over
associative semirings and their categories of \emph{semicomodules.} Our results
generalize old and recent results on corings over rings and their categories of
comodules. The generalization is \emph{not} straightforward and even subtle at
some places due to the nature of the base category of commutative monoids which
is neither Abelian (not even additive) nor homological, and has no non-zero
injective objects. To overcome these and other difficulties, a combination of
methods and techniques from categorical, homological and universal algebra is
used including a new notion of exact sequences of semimodules over semirings.
