A nonlinear version of resonance ultrasound spectroscopy (RUS) theory is presented. This is important for NDT-purposes as damage manifests itself more pronounced and in an earlier stage by changes in the nonlinear elastic constants. General equations are derived for the 1-D case, describing the interaction between the modes due to the presence of nonlinearity. An analytical solution of these equations is derived which predicts the shift of the resonance frequency versus amplitude in a bar with localized damage. The damage was modelled as a finite region, having a constant cubic nonlinearity, in an otherwise linear 1-D bar. The analytical expressions for the shifts in resonance frequency at different modes were used to infer information about the position, nonlinearity and width of the damage. Unlike other techniques, the proposed method does not require scanning to locate the defect, as it lets the different modes, each with a different vibration pattern, probe the structure.