  We study all possible deformations of the Maxwell algebra. In D=d+1\neq 3
dimensions there is only one-parameter deformation. The deformed algebra is
isomorphic to so(d+1,1)\oplus so(d,1) or to so(d,2)\oplus so(d,1) depending on
the signs of the deformation parameter. We construct in the dS(AdS) space a
model of massive particle interacting with Abelian vector field via non-local
Lorentz force. In D=2+1 the deformations depend on two parameters b and k. We
construct a phase diagram, with two parts of the (b,k) plane with so(3,1)\oplus
so(2,1) and so(2,2)\oplus so(2,1) algebras separated by a critical curve along
which the algebra is isomorphic to Iso(2,1)\oplus so(2,1). We introduce in
D=2+1 the Volkov-Akulov type model for a Abelian Goldstone-Nambu vector field
described by a non-linear action containing as its bilinear term the free
Chern-Simons Lagrangean.
