  We present an Eilenberg-Steenrod-like axiomatic framework for equivariant
coarse homology and cohomology theories. We also discuss a general construction
of such coarse theories from topological ones and the associated transgression
maps. A large part of this paper is devoted to showing how some
well-established coarse (co-)homology theories, whose equivariant versions are
either already known or will be introduced in this paper, fit into this setup.
Furthermore, a new and more flexible notion of coarse homotopy is given which
is more in the spirit of topological homotopies. Some, but not all, coarse
(co-)homology theories are even invariant under these new homotopies. They also
led us to a meaningful concept of topological actions of locally compact groups
on coarse spaces.
