In this study moving contact problem for a rigid cylindrical punch and a functionally graded layer is considered. The punch subjected to concentrated normal force, and moves steadily with a constant subsonic velocity on the boundary. Poisson's ratio is taken as constant, and both the elasticity modulus and the mass density are assumed to vary exponentially in depth direction. By using Fourier transform and boundary conditions, the governing equations are reduced to a Cauchy singular integral equation. The numerical solution of the singular integral equation is obtained by using Gauss-Chebyshev integration formulas. Numerical results for the contact area, the contact stress and the normal stresses are given. This study is limited in that the elasticity modulus and the mass density vary with the same function. However, it is the first attempt to investigate the moving contact problem with FGMs. (C) 2015 Elsevier Ltd. All rights reserved.