AbstractIn this article we introduce and study oscillating Gaussian processes defined by$$X_t = \alpha _+ Y_t \mathbf{1}_{Y_t >0} + \alpha _- Y_t\mathbf{1}_{Y_t<0}$$Xt=α+Yt1Yt>0+α-Yt1Yt<0, where$$\alpha _+,\alpha _->0$$α+,α->0are free parameters andYis either stationary or self-similar Gaussian process. We study the basic properties ofXand we consider estimation of the model parameters. In particular, we show that the moment estimators converge in$$L^p$$Lpand are, when suitably normalised, asymptotically normal.