  We describe an algorithm for computing Schur indices of irreducible
characters of a finite group $G$, based on computations within $G$ and its
subgroups and with their character tables. The algorithm has been implemented
within \Magma\ and examples of its use are given. We also compare the output of
the algorithm on groups associated with sporadic simple groups with the results
of W. Feit. The differences are considered.
