Viscoelasticity is exhibited by polymers, metals undergoing diffusion creep, etc. The strain is a linear functional of the stress, but there is no unique equilibrium relationship between them. Under a constant stress, the strain does not saturate to an equilibrium value. This divergence is the main difficulty facing a first-principles theory of viscoelasticity, in contrast to anelasticity which has already been understood as a relaxation process in terms of response theory, fluctuations and related concepts. We now present such a theory, based on the recognition that viscoelasticity occurs whenever the spontaneous fluctuations of the strainrate, butnot of the strain, form a stationary random process. We give fundamental formulas for the creep function and the complex compliance, in terms of the spontaneous fluctuations of the strain rate, that apply to both viscoelasticity and anelasticity. A detailed stochastic analysis of the basic viscoelastic network equation corroborates and complements these results. The unphysical instantaneous response of the network is eliminated, and the network parameters are related to internal fluctuations. A certain functional form of the creep function is derived that is common to several physical situations, a few of which are mentioned. Detailed applications will be taken up elsewhere.