This study investigates the stress wave propagation in circular aluminum cylinders bonded with a functionally graded adhesive layer subjected to an axial impulsive load. The adhesive joint consists of two identical (aluminum) cylinders and a functionally graded adhesive layer. The volume fractions of the two constituents: aluminum and epoxy in the adhesive layer were functionally tailored through the adhesive thickness by obeying a power-law. Therefore, the effective material properties at any point in the adhesive layer were predicted by the Mori-Tanaka homogenization scheme. The governing equations of the wave propagation in the joint were discretized by means of the finite difference method. The influence of the compositional gradient exponent on the displacement and stress distributions of the joint was examined. It was observed that changing the material composition of the adhesive layer had an evident effect on the displacement and stress levels, especially in the lower cylinder. On the contrary, the influence of the compositional gradient exponent was found to be minor on the displacement and stress distributions. The displacement and stress distributions were also investigated along the upper and lower cylinder-adhesive interfaces. Accordingly, with increasing the ductility of the adhesive layer the waves transmitted to the lower cylinder caused lower displacement levels. The normal stresses become peak at the bottom corners of the upper and lower cylinder-adhesive interfaces whereas the shear stresses concentrate in the middle region of the interfaces. In addition, the temporal variations of the displacement and stress components were evaluated at some critical points of the adhesive and lower cylinder. The compositional gradient exponent played an important role on the displacement and stress levels as well as the wave speeds in the adhesive and lower cylinder rather than in the upper cylinder. The stresses in the joints were observed to be alleviated by employing a functionally graded adhesive layer.