The subject of counting curves on 3-folds is extremely rich. Modern tools allow us to define several sophisticated invariants that we think of as "curve counts", such as Gromov-Witten, Donaldson-Thomas, Pandharipande-Thomas and Gopakumar-Vafa invariants. They are proven or expected to be equivalent to each other, in a precise sense. In this talk, I will give an idea of how these invariants are defined and what is known about them, focusing on these correspondences and rationality properties. Along the way, I'll explain a result of myself with Ivan Karpov concerning the rationality of PT generating series on Fano 3-folds; other than that, the talk will be mostly expository.