We show that new families of accelerating and almost nondiracting beams (solutions) for Maxwell's equations can be constructed. These are complex geometrical optics (CGO) solutions to Maxwell's equations with nonlinear limiting Carleman weights. They have the form of wave packets that propagate along circular trajectories while almost pre-serving a transverse intensity prole. We also show similar waves constructed using the approach combining CGO solutions and the Kelvin transform.