Self-polar polytopes are convex polytopes that are equal to an orthogonal
transformation of their polar sets. These polytopes were first studied by
Lov\'{a}sz as a means of establishing the chromatic number of distance graphs
on spheres, and they can also be used to construct triangle-free graphs with
arbitrarily high chromatic number. We investigate the existence, construction,
facial structure, and practical applications of self-polar polytopes, as well
as the place of these polytopes within the broader set of self-dual polytopes