In recent years, point-sampled geometry is becoming ubiquitous in graphics and geometric information processing. From a computer graphics point of view, the first major challenge in point-based geometry is to render high quality realistic images. A commonly used technique in realistic rendering is texture mapping, which essentially adds surface and/or material property detail in the final stages of rendering the image. The primary focus of the research reported in this thesis is to find suitable solutions to directly map one or more textures onto the surface of geometry represented by points without explicitly converting the surface to polygon mesh or to another geometric surface representation.  Parameterization is the most important step required for adding texture onto the surfaces of objects. In this thesis, a global parameterization method is developed by using level set methods to evolve the concerned surface to a surface with implied parameterization, say a sphere. By tracking the point samples to their final destinations on the sphere, a polar coordinate is assigned to every point in the original model. The user then chooses a few anchor points that map into one or more texture images to yield a simple and flexible procedure to map texture images onto the surface of a point-based model. The method has been implemented using MATLAB and C++ and tested on a number of point-based models