  It is shown that a constant magnetic field in 3+1 and 2+1 dimensions is a
strong catalyst of dynamical chiral symmetry breaking, leading to the
generation of a fermion dynamical mass even at the weakest attractive
interaction between fermions. The essence of this effect is the dimensional
reduction $D\to D-2$ in the dynamics of fermion pairing in a magnetic field.
The effect is illustrated in the Nambu--Jona--Lasinio (NJL) model and QED. In
the NJL model in a magnetic field, the low--energy effective action and the
spectrum of long wavelength collective excitations are derived. In QED (in
ladder and improved ladder approximations) the dynamical mass of fermions
(energy gap in the fermion spectrum) is determined. Possible applications of
this effect and its extension to inhomogeneous field configurations are
discussed.
