  We formulate the inverse spectral theory of infinite gap Hill's operators
with bounded periodic potential as a Riemann--Hilbert problem on a typically
infinite collection of spectral bands and gaps. We establish a uniqueness
theorem for this Riemann--Hilbert problem, which provides a new route to
establishing unique determination of periodic potentials from spectral data. As
the potential evolves according to the KdV equation, we use integrability to
derive an associated Riemann--Hilbert problem with explicit time dependence.
Basic principles from the theory of Riemann--Hilbert problems yield a new
characterization of spectra for periodic potentials in terms of the existence
of a solution to a scalar Riemann--Hilbert problem, and we derive a similar
condition on the spectrum for the temporal periodicity for an evolution under
the KdV equation.
