Many of the best studied types of tensor categories enjoy a notion of duality called rigidity (which among other nice properties implies that the tensor product is exact, if the underlying category is abelian). However, not all sources of interesting tensor categories obligingly produce rigid ones. In particular, the natural notion of duality for a category of modules over a vertex operator algebra is Grothendieck-Verdier duality, which will be the focus of this talk. I will discuss some recent results on tensor categories with Grothendieck-Verdier duality structures, module categories over these and why these results are encouraging for conformal field theory.