Integrable third order equations in 2+1 dimensions, which generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry Dym equations, will be classified. The approach is based on the observation that dispersionless limits of integrable systems in 2+1 dimensions possess infinitely many multi-phase solutions coming from the so-called hydrodynamic reductions.

Conversely, the requirement of the existence of hydrodynamic reductions proves to be an efficient classification criterion.

A novel perturbative approach to the classification problem will be adopted: based on the method of hydrodynamic reductions, one first classifies integrable quasi-linear systems, which may (potentially) occur as dispersionless limits of soliton equations in 2+1 dimensions.

To reconstruct dispersive deformations, one requires that all hydrodynamic reductions of the dispersionless limit are inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third order equations, some of which are apparently new.