  We introduce support varieties for rational representations of a linear
algebraic group $G$ of exponential type over an algebraically closed field $k$
of characteristic $p > 0$. These varieties are closed subspaces of the space
$V(G)$ of all 1-parameter subgroups of $G$. The functor $M \mapsto V(G)_M$
satisfies many of the standard properties of support varieties satisfied by
finite groups and other finite group schemes. Furthermore, there is a close
relationship between $V(G)_M$ and the family of support varieties $V_r(G)_M$
obtained by restricting the $G$ action to Frobenius kernels $G_{(r)} \subset
G$. These support varieties seem particularly appropriate for the investigation
of infinite dimensional rational $G$-modules.
