### Abstract

There is a classical result going at least as far back as Karlin and McGregor (around 1950) giving the transition probability p(x,y,t) or the N-step transition probability for a birth-and-death process in terms of some scalar valued orthogonal polynomials arising from the corresponding tridiagonal matrix. I will introduce the theory and some examples of matrix valued orthogonal polynomials, review the results of Karlin-McGregor, and show how these matrix valued polynomails allow one to deal with Markov chains that go beyond birth-and-death processes.