Summary: We solve a physically significant extension of a classic problem in the theory of diffusion, namely the Ornstein-Uhlenbeck process G. E. Uhlenbeck and L. S. Ornstein [Phys. Rev. (2) 36, 823–841 (1930; JFM 56.1277.03)]. Our generalized Ornstein-Uhlenbeck systems include a force which depends upon the position of the particle, as well as upon time. They exhibit anomalous diffusion at short times, and non-Maxwellian velocity distributions in equilibrium. Two approaches are used. Some statistics are obtained from a closed-form expression for the propagator of the Fokker-Planck equation for the case where the particle is initially at rest. In the general case we use spectral decomposition of a Fokker-Planck equation, employing nonlinear creation and annihilation operators to generate the spectrum which consists of two staggered ladders.