Solutions to the nonlinear Poisson-Boltzmann equation were used to obtain the electrostatic potentials of RNA molecules that have known three-dimensional structures. The results are described in terms of isopotential contours and surface electrostatic potential maps. Both representations have unexpected features: 'cavities' within isopotential contours and areas of enhanced negative potential on molecular surfaces. Intriguingly, the sites of unusual electrostatic features correspond to functionally important regions, suggesting that electrostatic properties play a key role in RNA recognition and stabilization. These calculations reveal that the electrostatic potentials generated by RNA molecules have a variety of functionally important characteristics that cannot be discerned by simple visual inspection of the molecular structure.