The adaptive high-gain output feedback strategy u(t) = -k (t)y(t), (d/dt)k(t) = parallel toy(t)parallel to(2) is, well established in the context of linear, minimum-phase, m-input m-output systems (A, B, C) with the property that spec(CB) C C+; the strategy applied to any such linear system achieves the performance objectives of: 1) global attractivity of the zero state and 2) convergence of the adapting gain to a finite limit. Here, these results are generalized in three aspects. First, the class of systems is enlarged to a class N-h(mu), encompassing nonlinear systems modeled by functional differential equations, where the parameter h greater than or equal to 0 quantifies system memory and the continuous function mu : [0, infinity) --> [0,infinity), with mu(0) = 0, relates to the allowable system nonlinearities. Second, the linear control law is replaced by u(t) = -k(t)[y(t) + mu(parallel toy(t)parallel to)/parallel toy(t)parallel to]y(t), wherein the additional nonlinear term counteracts the system nonlinearities. Third, the quadratic adaptation law is replaced by the law (d/dt) k (t) = psi(parallel toy(t)parallel to), where the continuous function psi satisfies certain growth conditions determined by mu (in particular cases, e.g., linear systems, a bounded function psi is admissible). Performance objectives 1) and 2) above are shown to persist in the generalized framework.