While the parameter convergence properties of standard adaptive algorithms for linear systems are well established, there are no similar results on the parameter convergence of adaptive controllers for nonlinear systems which have gained popularity in recent years, In this paper we focus on a recently developed class of adaptive schemes for output-feedback nonlinear systems and show that parameter convergence is guaranteed if and only if an appropriately defined signal vector, which does not depend on closed-loop signals, is persistently exciting, Then we develop an analytic procedure which allows us, given a specific nonlinear system and a specific reference signal, to determine a priori whether or not this vector is persistently exciting (PE) and, hence, whether or not the parameter estimates will converge. In the process we show that the presence of nonlinearities usually reduces the sufficient richness (SR) requirements on the reference signals and hence enhances parameter convergence, This is the first result on the relationship between PE and SR for adaptive nonlinear control systems.