Recently, a series of seminal results has established the central role that convex approximation plays in solving a number of computational problems in Euclidean spaces of constant dimension. These include fundamental applications such as computing the diameter of a point set, Euclidean minimum spanning trees, bichromatic closest pairs, width kernels, and nearest neighbor searching. In each case, an approximation parameter eps > 0 is given, and the problem is to compute a solution that is within a factor of 1+eps of the exact solution. In this talk, I will present recent approaches to convex approximation. These approaches achieve efficiency by being locally sensitive to the shape and curvature of the body. This is achieved through a combination of methods, both new and classical, including Delone sets in the Hilbert metric, Macbeath regions, and John ellipsoids. We will explore the development of these methods and discuss how they can be applied to the above problems.