  Some classes of the so called "travelling wave" solutions of Einstein and
Einstein - Maxwell equations in General Relativity and of dynamical equations
for massless bosonic fields in string gravity in four and higher dimensions are
presented. Similarly to the well known pp-waves, these travelling wave
solutions may depend on arbitrary functions of a null coordinate which
determine the arbitrary profiles and polarizations of the waves. However, in
contrast with pp-waves, these waves do not admit the null Killing vector fields
and can exist in some curved (expanding and spatially homogeneous) background
space-times, where these waves propagate in certain directions without any
scattering. Mathematically, some of these classes of solutions arise as the
fixed points of Kramer-Neugebauer transformations for hyperbolic integrable
reductions of the mentioned above field equations, or, in the other cases, --
after imposing of the ansatz that these waves do not change the part of spatial
metric transversal to the direction of wave propagation. It is worth to note
that strikingly simple forms of all presented solutions make possible a
consideration of nonlinear interaction of these waves with the background
curvature and singularities as well as a collision of sandwiches of such waves
with solitons or with each others in the backgrounds where such travelling
waves may exist.
