The main purpose of this article is to develop the generalized analytic Fourier--Feynman transform theory. We introduce a generalized analytic Fourier--Feynman transform and a multiple generalized analytic Fourier--Feynman transform with respect to Gaussian processes on the function space $C_{a,b}[0,T]$ induced by a generalized Brownian motion process. We then establish a relationship between these two generalized analytic transforms.