Integrable systems are dynamical systems which can in some sense be ‘solved explicitly’. The classification and deformations of 1 + 1 dimensional systems has been studied by B. Dubrovin, his collaborators and many others extensively. The study of 2 + 1 dimensional systems is ongoing and this thesis provides classification results in 2 + 1 dimensional systems based on the novel approach by the Loughborough group (E. Ferapontov, K. Khusnutdinova, V. Novikov). This involves firstly applying the method of hydrodynamic reductions to the dispersionless part of the systems and secondly reconstructing integrable dispersive systems by deforming the corresponding hydrodynamic reductions.