In this study, a frictional moving contact problem between a homogeneous magneto-electro-elastic half plane and a conducting rigid flat or cylindrical punch is considered. The punch moves on the half plane in the lateral direction at a subsonic constant velocity V. Applying the Fourier transform and Galilean transformation, the mixed boundary value problem is reduced to Cauchy integral equations in which the unknowns are the contact stress, the contact width, the electric displacement and the magnetic induction on the surface of the half plane. These singular integral equations are solved numerically using the Gauss-Jacobi and Gauss-Chebyshev integration formulas. Numerical results for the contact width, the contact stress, the in-plane stress, the electric displacement, and the magnetic induction on the surface of the half plane are obtained and plotted. This work is the first study that investigates the effect of friction in the moving contact problem of the magneto-electro-elastic half plane.