In recent years, the scanning of real-world physical objects using 3d acquisition devices has become an everyday task. Such 3d scanners output a set of three-dimensional points, called point cloud, sampling the surface of an object. These samples are typically subject to imperfections like noise, holes and outliers. In computer graphics, the field of surface reconstruction is concerned with the problem of converting this raw point data to a clean, more usable and visualizable representation. Some of the developed techniques aim at directly rendering a closed surface from such a point cloud, but also require certain precomputations to produce images of high quality. The ever-increasing acquisition rates and data throughputs of even low-cost scanner hardware have recently made it possible to capture dynamic and animated objects in real time. ^These high-speed acquisition capabilities call for new high-performance reconstruction algorithms that are able to keep up with these acquisition rates and allow an instant high-quality visualization of the real-time captured data. In this thesis, we pick up on this problem and develop various techniques that allow a fast processing and visualization of such raw, unstructured and potentially dynamic point clouds, which might be streamed to our computer at real-time rates from any possible source. In the course of our work, we investigate probabilistic methods that allow achieving a significant acceleration of state-of-the-art point-based operators, and use statistical models of 3d point sets to develop a fast technique for probabilistic surface reconstruction and representation. ^We develop a GPU point-rendering framework that performs any reconstruction computations required for a high-quality visualization instantly, i.e., on the fly at render time, and only on a necessary minimal subset of the data, i.e., the points visible on the screen. To this end, the first part of this thesis addresses a basic problem common to almost any surface-reconstruction technique, which is the fast and efficient search for spatial neighbors in an unstructured and unordered large collection of points. Knowing about a points nearest neighbors is an essential prerequisite for establishing local connectivity, assessing the shape of the surrounding surface, and applying filter operations for improving the quality of the geometric data and thus the resulting surface. ^In the second part, we improve on this direct reconstruction and rendering technique and present a more elaborate method that allows working at arbitrary reconstruction bandwidths, improves on the temporal stability of the rendered image, and produces a surface rendering of increased smoothness. In the third part, we focus on the problem of noise and outliers in the input data, and introduce a novel technique that allows for a fast feature-preserving resampling of unstructured dynamic point sets at render time. To this end, we describe the point cloud by a sparse probabilistic Gaussian mixture model, which allows for a much more compact representation and thus much faster operations on the spatial data. We will show that this technique significantly improves on the speed and even on the accuracy and quality of a feature-preserving point-set resampling operator. ^Based on the observed computational benefits of this probabilistic model, the final part of this thesis investigates a new way of defining a smooth and continuous surface solely based on a sparse Gaussian mixture. We will develop an entirely probabilistic reconstruction pipeline, and show that we can describe a feature-rich surface in a highly memory-efficient way while obtaining a reconstruction performance that can compete and even improve on the state of the art.