Convergence of the state errore to zero in adaptive systems is shown using the existence and uniqueness of solution and the existence of a Lyapunov function in which the adaptation laws are constructed. Results in the paper are general in the sense that it is applicable to a broad class of adaptive systems of a linear/nonlinear, time-varying or distributed-parameter systems. Since the approach taken in the paper does not require the boundedness of the derivative of the state errore for allt≥0, it is particularly useful in the adaptive control of infinite dimensional systems.