We study the shadows cast by the different types of rotating regular black holes viz. Ay\'on-Beato-Garc\'ia {(ABG)}, Hayward, and Bardeen. These black holes have in addition to the total mass ($M$) and rotation parameter ($a$), different parameters as electric charge ($Q$), deviation parameter ($g$), and magnetic charge ($g_{*}$), respectively. Interestingly, the size of the shadow is affected by these parameters in addition to the rotation parameter. We found that the radius of the shadow in each case decreases monotonically and the distortion parameter increases when the value of these parameters increase. A comparison with the standard Kerr case is also investigated. We have also studied the influence of the plasma environment around regular black holes to discuss its shadow. The presence of the plasma affects the apparent size of the regular black hole's shadow to be increased due to two effects (i) gravitational redshift of the photons and (ii) radial dependence of plasma density.Comment: 11 pages, 11 figures, accepted for publication in Physical Review

Based on the Newman-Janis algorithm the Ayón-Beato-García spacetime metric [1] of the regular spherically symmetric, static and charged black hole has been converted into rotational form. It is shown that the derived solution for rotating regular black hole is regular and the critical value of the electric charge Q for which two horizons merge into one sufficiently decreases in the presence of nonvanishing angular momentum a of the black hole.