  A family of explicit exact solutions of Einstein's equations in four and
higher dimensions is studied which describes photon rockets accelerating due to
an anisotropic emission of photons. It is possible to prescribe an arbitrary
motion, so that the acceleration of the rocket need not be uniform - both its
magnitude and direction may vary with time. Except at location of the
point-like rocket the spacetimes have no curvature singularities, and
topological defects like cosmic strings are also absent. Any value of a
cosmological constant is allowed. We investigate some particular examples of
motion, namely a straight flight and a circular trajectory, and we derive the
corresponding radiation patterns and the mass loss of the rockets. We also
demonstrate the absence of "gravitational aberration" in such spacetimes. This
interesting member of the higher-dimensional Robinson-Trautman class of pure
radiation spacetimes of algebraic type D generalises the class of Kinnersley's
solutions that has long been known in four-dimensional general relativity.
