  We demonstrate both theoretically and experimentally nonparaxial Mathieu and
Weber accelerating beams, generalizing the concept of previously found
accelerating beams. We show that such beams bend into large angles along
circular, elliptical or parabolic trajectories but still retain nondiffracting
and self-healing capabilities. The circular nonparaxial accelerating beams can
be considered as a special case of the Mathieu accelerating beams, while an
Airy beam is only a special case of the Weber beams at the paraxial limit. Not
only generalized nonparaxial accelerating beams open up many possibilities of
beam engineering for applications, but the fundamental concept developed here
can be applied to other linear wave systems in nature, ranging from
electromagnetic and elastic waves to matter waves.
