A sphere falling through a stable stratification of miscible fluids exhibits a prolonged settling rate due to the deformation of the fluid density field. A detailed experimental study is presented in which time-lapse images are analyzed to find the sphere and fluid interface positions. We additionally quantify the effect of the enhanced residence time through a settling-rate competition between spheres in homogeneous and stratified fluids. Various aspects of the experiments, such as convection, diffusion, and internal waves, are studied extensively. We use a Green's function formulation to derive from first principles a numerically assisted theoretical model for the behavior of the sphere and the surrounding fluid at low Reynolds number. This model is verified in a comparative study with experimental results from a wide range of parameters. Analysis of the theoretical model provides the streamlines and instantaneous stagnation points, affording some insight into the behavior of the interior of the fluid. The nondimensional form of the model is used to characterize the entire flow with only four parameters, and the impact of each of these parameters on the flow is studied numerically. With a model that shows good agreement with experimental data, we can then extend the theory to free space. In this regime, the formulation is exact, rather than an asymptotic approximation. We also discuss the convergence of volume integrals of the model in light of the Stokes paradox, in which a sphere in homogeneous fluid will drag an infinite volume. The model can be further pressed into a higher Reynolds number regime, which we then compare with experimental data. A brief look is taken at the extension to many-body sedimentation in which the similarities to a single settling sphere are noted. Additionally, we can approximate a linear stratification and find fairly good agreement.