Hybrid-based adaptive NN backstepping tracking control designs for both the single-input/single-output (SISO) and the square multi-input/multi-output (MIMO) strict-feedback systems with unknown system nonlinearities are presented. Each virtual/actual controller in these designs contains four main parts: a single-layer radial basis function neural network (RBFNN) for re-parameterizing the unknown nonlinearity to render the adaptive control applicable; an adaptive linearizing controller for compensating the resembled nonlinearities; a supervisory agent which hands over temporarily the control authority to the fourth part of a robust controller during the singularity. The proposed design ensures the semiglobal uniform ultimate boundedness (SGUUB) of all the closed-loop signals and compared with existing schemes has a wider applicability with a simpler structure. Simulation results demonstrating the validity of the proposed design are given in the final section. (C) 2009 Elsevier Ltd. All rights reserved.