Adaptive control of general nonlinear systems with nonlinear parameterization is studied in this paper. Under the assumptions that the system has stable free dynamics and satisfies controllability-like conditions characterized by the Lie brackets of affine vector fields, it is proved that there exist LgV-type adaptive controllers that not only asymptotically regulate the state of the nonlinearly parameterized system but also guarantee global stability of the closed-loop system. The design of LgV-type adaptive controllers is also included. Applications of the proposed adaptive control scheme are presented, including an interesting case of a dc-microgrid with photovoltaic (PV) and battery system.