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 a transition between them. 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
mention some of the corresponding results.