The notion of optimal polynomial approximant has appeared in the mathematics literature in the last few years in connection with Hilbert spaces of analytic functions of one or more variables and in the 1970s in connection with digital filter design. In this talk, I will examine the relationship between these polynomials and orthogonal polynomials as well as reproducing kernels in weighted spaces. I will describe what is known about the zeros of optimal polynomial approximants, their rates of decay, and convergence results, together with several open problems.