  We provide a semiclassical theory of tunneling decay in a magnetic field and
a three-dimensional potential of a general form. Because of broken
time-reversal symmetry, the standard WKB technique has to be modified. The
decay rate is found from the analysis of the set of the particle Hamiltonian
trajectories in complex phase space and time. In a magnetic field, the
tunneling particle comes out from the barrier with a finite velocity and behind
the boundary of the classically allowed region. The exit location is obtained
by matching the decaying and outgoing WKB waves at a caustic in complex
configuration space. Different branches of the WKB wave function match on the
switching surface in real space, where the slope of the wave function sharply
changes. The theory is not limited to tunneling from potential wells which are
parabolic near the minimum. For parabolic wells, we provide a bounce-type
formulation in a magnetic field. The theory is applied to specific models which
are relevant to tunneling from correlated two-dimensional electron systems in a
magnetic field parallel to the electron layer.
