The cosmological consequences of a pervasive, rolling, self-interacting, homogeneous scalar field are investigated. A number of models in which the energy density of the scalar field red-shifts in a specific manner are studied. In these models the current epoch is chosen to be scalar-field dominated to agree with dynamical estimates of the density parameter, Qd„„-0.2, and zero spatial curvature. The required scalar-field potential is "nonlinear" and decreases in magnitude as the value of the scalar field increases. A special solution of the field equations which is an attractive, timedependent, fixed point is presented. These models are consistent with the classical tests of gravitation theory. The Eotvos-Dicke measurements strongly constrain the coupling of the scalar field to light (nongravitationalj fields. Nucleosynthesis proceeds as in the standard hot big-bang model. In linear perturbation theory the behavior of baryonic perturbations, in the baryon-dominated epoch, do not differ significantly from the canonical scenario, while the presence of a substantial amount of homogeneous scalar-field energy density at low red-shifts inhibits the growth of perturbations in the baryonic fluid. The energy density in the scalar field is not appreciably perturbed by nonrelativistic gravitational fields, either in the radiation-dominated, matter-dominated, or scalar-field-dominated epochs. On the basis of this effect, we argue that these models could reconcile the low dynamical estimates of the mean mass density with the negligibly small spatial curvature preferred by inflation. One consequence of observational astronomy over the last half-century has been the accumulation of fairly persuasive evidence that a substantial fraction of the gravitationally bound mass associated with observed structure in the Universe is nonluminous. ' Dynamical estimates of the mass density on large (clusters of galaxies, etc. ) scales (which assumes that galaxies trace mass) suggest Qd~„— — 0.2+0. 1 (Ref. 2). [The density parameter Q is the ratio of the mean mass density to the Einstein — de Sitter value, Q(t) =8m.Gp/(3H ), where G is Newton's gravitational constant, H is the Hubble constant, and p the relevant mass density. ] The fact that the baryon mass density Qa needed for nucleosynthesis falls within this range is sometimes taken as evidence that the nonluminous mass is purely baryonic and that the total mass density in all forms, A,o, is appreciably less than the Einstein — de Sitter value. If the equations of general relativity (i.e., Einstein s theory of gravity with baryons and radiation) govern cosmology, this would mean that spatial sections must have appreciable mean curvature. However, this conflicts with the inflation paradigm,