In light of the complex and highly uncertain nature of dynamical systems requiring controls, it is not surprising that reliable system models for many high performance engineering and life science applications are unavailable. In the face of such high levels of system uncertainty, robust controllers may unnecessarily sacrifice system performance whereas adaptive controllers are clearly appropriate since they can tolerate far greater system uncertainty levels to improve system performance. In this dissertation, we develop a Lyapunov-based direct adaptive and neural adaptive control framework that addresses parametric uncertainty, unstructured uncertainty, disturbance rejection, amplitude and rate saturation constraints, and digital implementation issues. Specifically, we consider the following research topics: direct adaptive control for nonlinear uncertain systems with exogenous disturbances; robust adaptive control for nonlinear uncertain systems; adaptive control for nonlinear uncertain systems with actuator amplitude and rate saturation constraints; adaptive reduced-order dynamic compensation for nonlinear uncertain systems; direct adaptive control for nonlinear matrix second-order dynamical systems with state-dependent uncertainty; adaptive control for nonnegative and compartmental dynamical systems with applications to general anesthesia; direct adaptive control of nonnegative and compartmental dynamical systems with time delay; adaptive control for nonlinear nonnegative and compartmental dynamical systems with applications to clinical pharmacology; neural network adaptive control for nonlinear nonnegative dynamical systems; passivity-based neural network adaptive output feedback control for nonlinear nonnegative dynamical systems; neural network adaptive dynamic output feedback control for nonlinear nonnegative systems using tapped delay memory units; Lyapunov-based adaptive control framework for discrete-time nonlinear systems with exogenous disturbances; direct discrete-time adaptive control with guaranteed parameter error convergence; and hybrid adaptive control for nonlinear uncertain impulsive dynamical systems.