  Let G be a simply connected simple algebraic group over an algebraically
closed field K of characteristic p>0 with root system R, and let ${\mathfrak
g}={\cal L}(G)$ be its restricted Lie algebra. Let V be a finite dimensional
${\mathfrak g}$-module over K. For any point $v\inV$, the {\it isotropy
subalgebra} of $v$ in $\mathfrak g$ is ${\mathfrak g}_v=\{x\in{\mathfrak
g}/x\cdot v=0\}$. A restricted ${\mathfrak g}$-module V is called exceptional
if for each $v\in V$ the isotropy subalgebra ${\mathfrak g}_v$ contains a
non-central element (that is, ${\mathfrak g}_v\not\subseteq {\mathfrak
z(\mathfrak g)}$). This work is devoted to classifying irreducible exceptional
$\mathfrak g$-modules. A necessary condition for a $\mathfrak g$-module to be
exceptional is found and a complete classification of modules over groups of
exceptional type is obtained. For modules over groups of classical type, the
general problem is reduced to a short list of unclassified modules. The
classification of exceptional modules is expected to have applications in
modular invariant theory and in classifying modular simple Lie superalgebras.
