  For an $n\times n$ bounded matrix function $\Phi$ we study unitary
interpolants $U$, i.e., unitary-valued functions $U$ such that $\hat
U(j)=\hat\Phi(j)$, $j<0$. We are looking for unitary interpolants $U$ for which
the Toeplitz operator $T_U$ is Fredholm. We give a new approach based on
superoptimal singular values and thematic factorizations. We describe
Wiener--Hopf factorization indices of $U$ in terms of superoptimal singular
values of $\Phi$ and thematic indices of $\Phi-F$, where $F$ is a superoptimal
approximation of $\Phi$ by bounded analytic matrix functions. The approach
essentially relies on the notion of a monotone thematic factorization
introduced in [AP]. In the last section we discuss hereditary properties of
unitary interpolants. In particular, for matrix functions $\Phi$ of class
$H^\be+C$ we study unitary interpolants $U$ of class $QC$.
