The Markov sequence problem originates in work of Bochner concerning Markov processes on the sphere. The ultraspherical polynomials arise as eigenfunctions of certain Markov kernels. What sequences arise as the eigenvalue sequences of reversible Markov kernels having ultra spherical polynomials as their eigenfunctions? This question was answered by Bochner who described the extreme points of the convex set of such sequences. The problem has many connections with a wide range of problems in probability and analysis. This talk will present recent work growing out of a new approach to the Markov sequence problem by myself, Geronimo and Loss that originated in work on the Kac walk on high dimensional spheres. We present simple new proofs of classical results of Gasper for Jacobi polynomials, and of recent results of Griffith and Diaconis on Krawtchouck polynomials by Miclo using an extension of the method of myself Geronimo and Loss, as well as several new results.