A support vector machines (SVM)-based two-stage method is proposed to simulate and predict the nonlinear dynamic response of structures. In the first stage, an autoregressive moving average with exogenous input (ARMAX) model is used to represent the acceleration response as the output of a single-input single-output (SISO) system and the least square method is used to estimate the model parameters with which the linear acceleration response of the system can be simulated and predicted. Then the linear velocity and displacement are estimated using numerical integration of the predicted acceleration. In the second stage, by using the predicted linear responses (acceleration, velocity and displacement) and the excitation to construct the input vector, the SVM is used to approximate nonlinear mapping from the input vector to system output and the trained SVM can be used to simulate and predict the nonlinear dynamic response conveniently. The nonlinear dynamic responses of a Duffing oscillator and a frame structure are simulated and predicted using the proposed method as well as the neural network-based method. The results demonstrate that the SVM-based method provides superior performance in generalization and accuracy and can be a powerful tool for nonlinear system simulation and prediction.