We study the Faber - Krahn inequality for the Dirichlet eigenvalue problem of the 
Laplacian, first in $R^N$ , then on a compact smooth Riemannian manifold M . For the 
latter, we consider two cases. In the first case, the compact manifold has a lower bound 
on the Ricci curvature, in the second, the integral of the reciprocal of an isoperimetric 
estimator function of the Riemannian manifold is convergent. In all cases, we show 
that the first eigenvalue of a domain in $R^N$ , respectively M , is minimal for the ball 
of the same volume, respectively, for a geodesic ball of the same relative volume in 
an appropriate manifold $M^∗$ . While working with the isoperimetric estimator, the 
manifold $M^∗$ need not have constant sectional curvature. In $R^N$ , we also consider the 
Neumann eigenvalue problem and present the Szeg¨ 
o - Weinberger inequality. In this 
case, the principal eigenvalue of the ball is maximal among all principal eigenvalues 
of domains with same volume.