Persistence modules are the algebraic objects at the heart of various theories of persistence. I will discuss some recent work with Bubenik and Scott on generalised persistence modules (viewed categorically), as well as some other studies with Chazal, Oudot and Glisse on the structure of 1-parameter persistence modules (the classical case). The idea is to move beyond the usual restriction to modules with "finitely many critical points" to allow much broader families of examples.