We introduce the Fukaya category of a symplectic manifold, with a particular focus on the case of surfaces (where many of definitions can be made combinatorial). In this case the objects of the Fukaya category are (decorated) immersed curves in the surface and morphisms count pseudoholomorphic disks. We also survey Heegaard Floer homology as well as its variants knot Floer homology and bordered Floer homology and related invariants such as Khovanov homology. We highlight some recent applications of the interplay between immersed curves in surfaces and these 3-dimensional invariants.