The responses of linear and nonlinear oscillators to fractional Gaussian noise (fGn) are studied. First, some preliminary concepts and properties of fractional Brownian motion (fBm) and fGn with Hurst index are introduced. Then, the exact sample solution, correlation function, spectral density, and mean-square value of the response of linear oscillator to fGn are obtained. Based on the sample solution, it is proved that the long-range correlation index of displacement response of linear oscillator is the same as that of excitation fGn, i.e., , while the velocity response has no such long-range correlation. An interesting discovery is that the ratio of kinetic energy to total energy decreases as increasing Hurst index . Finally, for the responses of one and two degrees-of-freedom (DOF) nonlinear oscillators to fGn, the equivalent linearization method is applied to obtain the sample functions, correlation functions and mean-square values of the responses. Plenty of digital simulation results are obtained to support these solutions. It is shown that the approximate solution is effective for weakly nonlinear oscillators and it is feasible to apply the equivalent linearization to study multi-DOF weakly nonlinear oscillators.