  We present a general theory of three-dimensional nonparaxial
spatially-accelerating waves of the Maxwell equations. These waves constitute a
two-dimensional structure exhibiting shape-invariant propagation along
semicircular trajectories. We provide classification and characterization of
possible shapes of such beams, expressed through the angular spectra of
parabolic, oblate and prolate spheroidal fields. Our results facilitate the
design of accelerating beams with novel structures, broadening scope and
potential applications of accelerating beams.
