  We study exact solutions of Dirac and Klein-Gordon equations and Green
functions in d-dimensional QED and in an external electromagnetic field with
constant and homogeneous field invariants. The cases of even and odd dimensions
are considered separately, they are essentially different. We direct attention
to the asymmetry of the quasienergy spectrum, which appears in odd dimensions.
The in and out classification of the exact solutions as well as the
completeness and orthogonality relations is strictly proven. Different Green
functions in the form of sums over the exact solutions are constructed. The
Fock-Schwinger proper time integral representations of these Green functions
are found. As physical applications we consider the calculations of different
quantum effects related to the vacuum instability in the external field. For
example, we present mean values of particles created from the vacuum, the
probability of the vacuum remaining a vacuum, the effective action, and the
expectation values of the current and energy-momentum tensor.
