This is joint work with Alexander Its and Igor Krasovsky. We study asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general non-degenerate behavior 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 sysem of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.