Abstarct: The conveying fluid pipes are regarded as the simplest system of fluid–structure interaction problems, which is capable of displaying very rich and interesting dynamical characteristics. Pipeline systems are used for a fluid conveyance are not only commonly found in oil and gas installations, pump discharge lines and heat exchangers, but also MEMS, NEMS and so on. Dynamic behavior of both simply support and cantilever functionally graded of the conveying fluid pipes, are studied. Two cases are considered for each boundary condition: A) pipe is functionally graded in thickness direction. B) pipe is functionally graded in its axial direction. Material graduation in axially or transversally through the thickness baxsed on the power law, and the motion equations of system are derived by using the principle of virtual work under the assumptions of the Euler–Bernoulli beam theory. The natural frequencies of conveing fluid pipe are obtained as the varius dimensionless parameters, and the effect of the gradual strengthening of the pipe is studied. The critical flow velocities which cause the unstablity of the system are obtained. It was found that in both types of boundary conditions, in the case of the Young's modulus while the length of the pipe gradually varies, by increasing the Young's modulus in the side of flow entrance, natural frequencies of pipe and the critical fluid velocity increases. Although, in the case of the Young's modulus of the pipe varies in the thickness direction, with growth in the outer surfaces of the pipe, the natural frequencies and the critical fluid velocity increases, and vice versa.