  We construct regular rotating black hole and no-horizon spacetimes based on
the recently introduced spherically symmetric generic regular black hole
spacetimes related to electric or magnetic charge under nonlinear
electrodynamics coupled to general relativity that for special values of the
spacetime parameters reduce to the Bardeen and Hayward spacetimes. We show that
the weak and strong energy conditions are violated inside the Cauchy horizons
of these generic rotating black holes. We give the boundary between the
rotating black hole and no-horizon spacetimes and determine the black hole
horizons and the boundary of the ergosphere. We introduce the separated Carter
equations for the geodesic motion in these rotating spacetimes. For the most
interesting new class of the regular spacetimes, corresponding for magnetic
charges to the Maxwell field in the weak field limit of the nonlinear
electrodynamics, we determine the structure of the circular geodesics and
discuss their properties. We study the epicyclic motion of a neutral particle
moving along the stable circular orbits around the "Maxwellian" rotating
regular black holes. We show that epicyclic frequencies measured by the distant
observers and related to the oscillatory motion of the neutral test particle
along the stable circular orbits around the rotating singular and regular
Maxwellian black holes are always smaller than ones in the Kerr spacetime.
