In this paper, we consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and, in particular, manifolds with nonnegative Ricci curvature. Busemann functions, which are the limits of distance functions about points approaching infinity, have been used in the past by S.T.Yau to prove that noncompact manifolds with nonnegative Ricci curvature have at least linear volume growth. We first prove a volume comparison theorem for sets covered by the flows of the gradient of a Busemann function. We then apply this theorem to study manifolds with the given lower Ricci curvature bounds and minimal volume growth, in particular, manifolds with nonnegative Ricci curvature and linear volume growth. We prove that the level sets of Busemann functions in such manifolds are compact and the diameters of the levels grow at most linearly. Thus a Busemann function is an exhaustion. We close with examples of manifolds in which the diameter of a Busemann function grows logarithmically. These constructions have metrics of doubly-warped product type for which the warping functions are solutions to an integrable ordinary differential equation.