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.