I will present the general construction of lattice homology. It is a bigraded Z[U] module. Associated with the topological type of a complex normal surface singularity, it categorifies the Seiberg Witten invariant of the link.
Its analytic version associated with a surface singularity categorifies the geometric genus (while the Euler characteristic in the curve case is the delta invariant).
I also present a different version, which can be applied for certain Artinian algebras. In this case it categorifies the dimension of the algebra.