  We review recent results on necessary and sufficient conditions for measures
on $\mathbb{R}$ and $\partial\mathbb{D}$ to yield exponential decay of the
recursion coefficients of the corresponding orthogonal polynomials. We include
results on the relation of detailed asymptotics of the recursion coefficients
to detailed analyticity of the measures. We present an analog of Carmona's
formula for OPRL. A major role is played by the Szego and Jost functions.
