  Let $\Phi$ be a nuclear space and let $\Phi'_{\beta}$ denote its strong dual.
In this work we establish the one-to-one correspondence between infinitely
divisible measures on $\Phi'_{\beta}$ and L\'{e}vy processes taking values in
$\Phi'_{\beta}$. Moreover, we prove the L\'{e}vy-It\^{o} decomposition, the
L\'{e}vy-Khintchine formula and the existence of c\`{a}dl\`{a}g versions for
$\Phi'_{\beta}$-valued L\'{e}vy processes. A characterization for L\'{e}vy
measures on $\Phi'_{\beta}$ is also established. Finally, we prove the
L\'{e}vy-Khintchine formula for infinitely divisible measures on
$\Phi'_{\beta}$.
