  In this survey we show how to produce asymptotics of determinants of
structured matrices using operator theory methods. We describe the asymptotics
for finite Toeplitz matrices, finite Toeplitz plus Hankel matrices and
generalizations of these, and also finite sections of functions of Toeplitz
operators, all with smooth matrix-valued symbols. Many of these results are
well-known. However, the results for certain Toeplitz plus Hankel operators
with matrix-valued symbol are presented for the first time and the result for
the finite sections of functions of Toeplitz operators is new.
