  In this paper, we prove a generalization of Reilly's formula in
\cite{Reilly}. We apply such general Reilly's formula to give alternative
proofs of the Alexandrov's Theorem and the Heintze-Karcher inequality in the
hemisphere and in the hyperbolic space. Moreover, we use the general Reilly's
formula to prove a new Heintze-Karcher inequality for Riemannian manifolds with
boundary and sectional curvature bounded below.
