  We present a number of D=4 bosonic and heterotic string solutions with a
covariantly constant null Killing vector which, like the solution of Nappi and
Witten (NW), correspond to (gauged) WZW models and thus have a direct conformal
field theory interpretation. A class of exact plane wave solutions (which
includes the NW solution) is constructed by `boosting' the twisted products of
two D=2 `cosmological' or `black-hole' cosets of SL(2,R) and SU(2). We describe
a general limiting procedure by which one can construct new solutions with a
covariantly constant null Killing vector starting with known string
backgrounds. By applying a non-abelian duality transformation to the NW model
we find a D=4 solution which has a covariantly constant null Killing vector but
is not a plane wave. Higher dimensional bosonic backgrounds with isometries can
be interpreted as lower dimensional backgrounds with extra gauge fields. Some
of them are at the same time solutions of the heterotic string theory. In
particular, the NW model represents also a D=3 `gravi-electromagnetic'
heterotic string plane wave. In addition to the (1,1) supersymmetric embeddings
of bosonic solutions we construct a number of non-trivial (1,0) supersymmetric
exact D=4 heterotic string plane wave solutions some of which are related (by a
boost and analytic continuation) to limiting cases of D=4 heterotic black hole
solutions.
