It is known that an anti-plane shear surface wave does not exist in anisotropic elastic half-space x 2 ⩾ 0 when the material is homogeneous. For a functionally graded material for which the elastic stiffness C 44, C 45, C 55 and the mass density ρ depend on the depth x 2 of the half-space, an anti-plane shear surface wave may exist. Exact solutions for four cases of special graded materials are presented here. In all four cases C 44 and ρ have the same function form. C 55 and C 45 are related but they need not have the same function form as C 44 and ρ. In fact C 45 can depend on x 2 arbitrarily. In Case 4, C 44 and ρ are periodic function in x 2. For a general graded material, an asymptotic solution for large wave number k is obtained and conditions for the existence of an anti-plane shear surface wave are presented. Regardless of exact or asymptotic solutions, as k → ∞, the surface wave speed v approaches v ˆ which is the speed of the exceptional body wave evaluated for the material property at x 2 = 0. The surface wave solution has an interesting phenomenon when the elastic stiffness C 45 is non-zero.