Since existing adaptive observers for nonlinear systems may generate unbounded parameter estimates in the presence of bounded disturbances, robust adaptive observers are presented which prevent parameter estimate drift. In addition the input to state stability of the error dynamics with respect to disturbances and parameter time-derivatives is guaranteed by generalizing a persistency of excitation result. Asymptotic convergence of state estimation errors is still achieved for systems in adaptive observer form when disturbances are not present, by a suitable extension of Barbalat's Lemma.