In this study, frictional moving contact problem for a rigid cylindrical punch and an elastic layer is considered. The punch is subjected to concentrated normal and tangential force, and moves steadily with a constant subsonic velocity on the boundary. The problem is reduced to a singular integral equation of the second kind, in which the contact stress and the contact area are the unknowns, and it is treated using Fourier transforms and the boundary conditions for the problem. The numerical solution of the singular integral equation is obtained by using the Gauss-Jacobi integration formulas. Numerical results for the contact stress and the contact area are given. The results show that with increasing values of relative moving velocity, contact width between the moving punch and the layer increases, whereas contact stress decreases.