We study the growth of the Mordell-Weil groups $E(K_n)$ of an elliptic curve $E$ as $K_n$ runs through the intermediate fields of a $\Bbb Z_p$-extension. We describe those $\Bbb Z_p$-extensions $K_\infty/K$ where we expect the ranks to grow to infinity. In the cases where we know or expect the rank to grow, we discuss where we expect to find the submodule of universal norms.
