  In 1976 Lieb and Thirring established upper bounds on sums of powers of the
negative eigenvalues of a Schr\"odinger operator in terms of semiclassical
phase-space integrals. Over the last 45 years the optimal constants in these
inequalities, the values of which were conjectured by Lieb and Thirring, have
been subject of intense investigations. We aim to review existing results.
