The medial axis transform of an object is a representation of a closed object as a union of the maximal inscribed balls interior to the object. It is a powerful shape descriptor with numerous applications in computer vision, computer graphics, medical imaging, robotics and GIS. I will describe a method for approximating the medial axis transform of a 3D solid with a dense collection of points near the salient parts of the medial axis. Correctness issues of the algorithm will be discussed. Further, I will show how this approximation to the medial axis transform faithfully captures information about the differential geometry of the solid boundary. Next, I will describe how our approximation of a solid using a small number of medial spheres offers a useful representation for accelerating proximity queries between solids. I will conclude by discussing how the medial axis of 2D shapes can be useful for cartographic generalization.