  In the last few years, the notion of optimal polynomial approximant has
appeared in the mathematics literature in connection with Hilbert spaces of
analytic functions of one or more variables. In the 70s, researchers in
engineering and applied mathematics introduced least-squares inverses in the
context of digital filters in signal processing. It turns out that in the Hardy
space $H^2$ these objects are identical. This paper is a survey of known
results about optimal polynomial approximants. In particular, we will examine
their connections with orthogonal polynomials and reproducing kernels in
weighted spaces and digital filter design. We will also describe what is known
about the zeros of optimal polynomial approximants, their rates of decay, and
convergence results. Throughout the paper, we state many open questions that
may be of interest.
