Universal vector wave equations allowing for a unified description of anyons and also of usual bosons and fermions in the plane are proposed The existence of two essentially different types of anyons based on unitary and also on non-unitary infinite-dimensional half bounded representations of the (2+1)D Lorentz algebra is revealed Those associated with non-unitary representations interpolate between bosons and fermions The extended formulation of the theory includes the previously known Jackiw-Nair UN) and MaJorana-Dirac (MD) descriptions of anyons as particular cases and allows us to compose bosons and fermions from entangled anyons The theory admits a simple supersymmetric generalization in which the JN and MD systems are unified in N=1 and N=2 supermultiplets Two different non-relativistic limits of the theory are investigated The usual one generalizes Levy-Leblond s spin 1/2 theory to arbitrary spin as well as to anyons The second Jackiw-Nair limit (that corresponds to Inonu-Wigner contraction with both anyon spin and light velocity going to infinity) is generalized to boson/fermion fields and interpolating anyons The resulting exotic Galilei symmetry is studied in both the non-supersymmetric and supersymmetric cases (C) 2010 Elsevier Inc All rights reserved