In this paper, we design adaptive observers for nonlinear systems in adaptive observer forms. These observers achieve not only asymptotical but also finite-time convergence in the observation of the state variables as well as the adaptation of the unknown parameters, under the normal persistence excitation condition. Proofs of the convergence are given by the local finite-time stability and by introducing high gains to dominate the unknown parameters. The design methods are applied for a robot manipulator. Simulation results show effectiveness of our new methods.