An analytical solution is obtained for functionally graded piezoelectric cylindrical structures, and the electroelastic behavior is investigated. The material properties are assumed to vary as an exponential function along the radial direction of the cylinder. The cylindrical structure acts as sensor or actuator and is subjected to mechanical and electrical loads at its inner and outer cylindrical surfaces. Based on the linear piezoelectricity theory, the governing equation in terms of the radial displacement is obtained as a second-order nonhomogeneous ordinary differential equation with variable coefficients, and its solution is obtained by the Frobenius series method. The illustrative examples show that the material nonhomogeneity has significant effect on the electroelastic fields of functionally graded piezoelectric cylindrical sensors and actuators.