In this talk I will discuss how to construct generalized traces on an ideal in certain module categories. As I will explain there are several examples in representation theory where the usual trace and dimension are zero, but these generalized traces and modified dimensions are non-zero. Such examples include the representation theory of the Lie algebra sl(2) over a field of positive characteristic and quantum groups. I will also explain how traces give rise to topological invariants of links. This is joint work with Jon Kujawa, Bertrand Patureau, Vladimir Turaev and Alexis Virelizier.