  In this paper, we use our previous study of the higher order Bernoulli
numbers $B_n^{(l)}$ to investigate the $p$-adic properties of the Stirling
numbers of the second kind $S(n,k)$. For example, we give a new, greatly
simplified proof of the formula $\nu_2(S(2^h,k))=d_2(k)-1$ if $1\le k \le 2^h$,
and generalize this result to arbitrary primes $p$. We also consider the
Stirling numbers of the first kind $s(n,k)$, with new results analogous to
those for the Stirling numbers of the second kind. New mod $p$ congruences for
Stirling numbers of both kinds are also given.
