  In this survey article, we review some results and conjectures related to
orthogonal polynomials on Cantor sets. The main purpose of this paper is to
emphasize the role of equilibrium measures in order to have a general theory of
sufficiently good measures (measures that behave similarly to measures which
are in the Szeg\H{o} class and the isospectral torus in the finite gap case)
supported on totally disconnected subsets of $\mathbb{R}$. We present some open
problems a number of which can be studied numerically.
