In this talk, I will present the theory of homological link
invariants, together with some recent developments. The theory
consists in categorifying various polynomial link invariants - like
Jones, HOMFLYPT or Alexander polynomial - by defining the graded chain
complexes whose homotopy type is the invariant of a link, and whose
graded Euler characterstic is equal to the polynomial link invariants.
I will show some basic constructions for the Jones polynomial
(sl(2)-link invariant), so-called Khovanov homology, as well as its
generalization for the HOMFLYPT polynomial (sl(n)-link invariant),
so-called Khovano-Rozansky homology, and also some recent
constructions of these homologies. If time permits, I will try to
explain the knot Floer homology by Ozsvath and Szabo, which
categorifies the Alexander polynomial.