A novel robust adaptive neural network (NN) control scheme with prescribed performance is developed for the 3-D trajectory tracking of underactuated autonomous underwater vehicles (AUVs) with uncertain dynamics and unknown disturbances using new prescribed performance functions, an additional term, the radial basis function (RBF) NN, and the command-filtered backstepping approach. Different from the traditional prescribed performance functions, the new prescribed performance functions are innovatively proposed such that the time desired for the trajectory tracking errors of AUVs to reach and stay within the prescribed error tolerance band can be preset exactly and flexibly. The additional term with the Nussbaum function is designed to deal with the underactuation problem of AUVs. By means of RBF NN, the uncertain item lumped by the uncertain dynamics of AUVs and unknown disturbances is eventually transformed into a linearly parametric form with only a single unknown parameter. The developed control scheme ensures that all signals in the AUV 3-D trajectory tracking closed-loop control system are bounded. Simulation results with comparisons show the validity and the superiority of our developed control scheme.