  Eigenvalue estimates that are optimal in some sense have self-evident appeal
and leave estimators with a sense of virtue and economy. So, it is natural that
ongoing searches for effective strategies for difficult tasks such as
estimating matrix eigenvalues that are situated well into the interior of the
spectrum revisit from time to time methods that are known to yield optimal
bounds. This article reviews a variety of results related to obtaining optimal
bounds to matrix eigenvalues --- some results are well-known; others are less
known; and a few are new. We focus especially on Ritz and harmonic Ritz values,
and right- and left-definite variants of Lehmann's method.
