  The propagator of a spinning particle in external Abelian field and in
arbitrary dimensions is presented by means of a path integral. The problem has
different solutions in even and odd dimensions. In even dimensions the
representation is just a generalization of one in four dimensions (it has been
known before). In this case a gauge invariant part of the effective action in
the path integral has a form of the standard (Berezin-Marinov) pseudoclassical
action. In odd dimensions the solution is presented for the first time and, in
particular, it turns out that the gauge invariant part of the effective action
differs from the standard one. We propose this new action as a candidate to
describe spinning particles in odd dimensions. Studying the hamiltonization of
the pseudoclassical theory with this action, we show that the operator
quantization leads to adequate minimal quantum theory of spinning particles in
odd dimensions. In contrast with the models proposed formerly in this case the
new one admits both the operator and the path integral quantization. Finally
the consideration is generalized for the case of the particle with anomalous
magnetic moment.
