  We present exact non-singular bounce solutions of general relativity in the
presence of a positive cosmological constant and an electromagnetic field,
without any exotic matter. The solutions are distinguished by being spatially
inhomogeneous in one direction, while they can also contain non-trivial
electromagnetic field lines. The inhomogeneity may be substantial, for
instance, there can be one bounce in one region of the universe and two bounces
elsewhere. Since the bounces are followed by a phase of accelerated expansion,
the metrics described here also permit the study of (geodesically complete)
models of inflation with inhomogeneous initial conditions. Our solutions admit
two Killing vectors and may be re-interpreted as the pathology-free interior
regions of Kerr-de Sitter black holes with non-trivial NUT charge. Remarkably
enough, within this cosmological context, the NUT parameter does not introduce
any string singularity nor closed timelike curves but renders the geometry
everywhere regular, eliminating the big bang singularity by means of a bounce.
