  We review the asymptotic behavior of a class of Toeplitz (as well as related
Hankel and Toeplitz + Hankel) determinants which arise in integrable models and
other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and how a
transition between them is related to the Painleve V equation. Certain Toeplitz
and Hankel determinants reduce, in certain double-scaling limits, to Fredholm
determinants which appear in the theory of group representations, in random
matrices, random permutations and partitions. The connection to Toeplitz
determinants helps to evaluate the asymptotics of related Fredholm determinants
in situations of interest, and we review the corresponding results.
