  This manuscript was written for the Proceedings of the ICRA 2022 in Buenos
Aires. It can be divided into four parts: The first part is an introduction to
the theory of monomorphism categories, including a short survey on some
representation theoretic results. The second part is a summary of some recent
results on monomorphism categories based on joint work with Nan Gao, Julian
K\"ulshammer and Chrysostomos Psaroudakis. It also includes a new result,
namely a classification of all monomorphism categories of finite type over
cyclic abelian groups. The third part concerns the relationship between
submodule categories and $p$-valuated abelian groups. The last part contains a
proof of wildness of a certain monomorphism category, rectifying a statement in
the literature.
