This thesis investigates symmetric Feller processes on uniform state spaces equipped with a measure, aiming to extend and unify diverse results on the convergence of such processes on metric measure spaces. It has been shown in previous works that processes that are related to both the metric and the measure of their state spaces through their Dirichlet forms converge whenever the state spaces converge. However, it is not always clear which metric is the right one to consider and there might even be many such metrics. The main focus lies therefore on abstracting from the metric structure of the spaces and instead considering their uniform structure. This approach warrants an in-depth analysis of uniform spaces as state spaces for Feller processes including an analysis of the Skorokhod topology on the the space of paths on such spaces. One of the main results is that the convergence of a family of hitting times implies the convergence of paths. Moreover, symmetric Feller processes on uniform state spaces and their Dirichlet forms are introduced and studied. As a result of a detailed study of killed processes, it is demonstrated that symmetric Feller processes are uniquely determined by their Green operators. Finally, five conditions for the convergence of symmetric Feller processes on uniform state spaces are identified and discussed.