We study the asymptotics in n for n-dimensional Toeplitz determinants whose
symbols possess Fisher-Hartwig singularities on a smooth background. We prove
the general non-degenerate asymptotic behavior as conjectured by Basor and
Tracy. We also obtain asymptotics of Hankel determinants on a finite interval
as well as determinants of Toeplitz+Hankel type. Our analysis is based on a
study of the related system of orthogonal polynomials on the unit circle using
the Riemann-Hilbert approach.