  We consider Chebyshev polynomials, $T_n(z)$, for infinite, compact sets
$\frak{e} \subset \mathbb{R}$ (that is, the monic polynomials minimizing the
sup-norm, $\Vert T_n \Vert_{\frak{e}}$, on $\frak{e}$). We resolve a $45+$ year
old conjecture of Widom that for finite gap subsets of $\mathbb{R}$, his
conjectured asymptotics (which we call Szeg\H{o}-Widom asymptotics) holds. We
also prove the first upper bounds of the form $\Vert T_n \Vert_{\frak{e}} \leq
Q C({\frak{e}})^n$ (where $C(\frak{e})$ is the logarithmic capacity of
$\frak{e}$) for a class of $\frak{e}$'s with an infinite number of components,
explicitly for those $\frak{e} \subset \mathbb{R}$ that obey a Parreau-Widom
condition.
