Summary: The monomorphism category \(\mathcal{M}(A, M, B)\) induced by a bimodule \({}_AM_B\) is the subcategory of {\(\Lambda\)}-mod consisting of \(\left[\begin{matrix} X \\ Y \end{matrix}\right]_\phi\) such that \(\phi : M \otimes_B Y \rightarrow X\) is a monic \(A\)-map, where \({\Lambda} = \left[\begin{matrix} A & M \\ 0 & B \end{matrix}\right]\), and \(A\), \(B\) are Artin algebras. In general, \(\mathcal{M}(A, M, B)\) is not the monomorphism category induced by quivers. It could describe the Gorenstein-projective \(\Lambda\)-modules. This monomorphism category is a resolving subcategory of \(\Lambda\)-mod if and only if \(M_B\) is projective. In this case, it has enough injective objects and Auslander-Reiten sequences, and can be also described as the left perpendicular category of a unique basic cotilting \(\Lambda\)-module. If \(M\) satisfies the condition (IP) (see Subsection 1.6), then the stable category of \(\mathcal{M}(A, M, B)\) admits a recollement of additive categories, which is in fact a recollement of singularity categories if \(\mathcal{M}(A, M, B)\) is a Frobenius category. Ringel-Schmidmeier-Simson equivalence between \(\mathcal{M}(A, M, B)\) and its dual is introduced. If \(M\) is an exchangeable bimodule, then an RSS equivalence is given by a \(\Lambda-\Lambda\) bimodule which is a two-sided cotilting \(\Lambda\)-module with a special property; and the Nakayama functor \(\mathcal{N}_{\Lambda}\) gives an RSS equivalence if and only if both \(A\) and \(B\) are Frobenius algebras.