  We study Galois points for a hypersurface $X$ with $\dim {\rm Sing}(X) \le
\dim X-2$. The purpose of this article is to determine the set $\Delta(X)$ of
Galois points in characteristic zero: Indeed, we give a sharp upper bound of
the number of Galois points in terms of $\dim X$ and $\dim {\rm Sing}(X)$ if
$\Delta(X)$ is a finite set, and prove that $X$ is a cone if $\Delta(X)$ is
infinite. To achieve our purpose, we need a certain hyperplane section theorem
on Galois point. We prove this theorem in arbitrary characteristic. On the
other hand, the hyperplane section theorem has other important applications:
For example, we can classify the Galois group induced from a Galois point in
arbitrary characteristic and determine the distribution of Galois points for a
Fermat hypersurface of degree $p^e+1$ in characteristic $p>0$.
