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 results on the sine, Bessel, and confluent hypergeometric kernel determinants. The talk is based on the joint works with Tom Claeys, Percy Deift, Alexander Its, and Julia Vasilevska.