  A particular yet large class of non-diverging solutions which admits a
cosmological constant, electromagnetic field, pure radiation and/or general
non-null matter component is explicitly presented. These spacetimes represent
exact gravitational waves of arbitrary profiles which propagate in background
universes such as Minkowski, conformally flat (anti-)de Sitter, Edgar-Ludwig,
Bertotti-Robinson, and type D (anti-)Nariai or Plebanski-Hacyan spaces, and
their generalizations. All possibilities are discussed and are interpreted
using a unifying simple metric form. Sandwich and impulsive waves propagating
in the above background spaces with different geometries and matter content can
easily be constructed. New solutions are identified, e.g. type D pure radiation
or explicit type II electrovacuum waves in (anti-)Nariai universe. It is also
shown that, in general, there are no conformally flat Einstein-Maxwell fields
with a non-vanishing cosmological constant.
