  A $\Sigma^*$-algebra is a concrete $C^*$-algebra that is sequentially closed
in the weak operator topology. We study an appropriate class of $C^*$-modules
over $\Sigma^*$-algebras analogous to the class of $W^*$-modules (selfdual
$C^*$-modules over $W^*$-algebras), and we are able to obtain
$\Sigma^*$-versions of virtually all the results in the basic theory of $C^*$-
and $W^*$-modules. In the second half of the paper, we study modules possessing
a weak sequential form of the condition of being countably generated. A
particular highlight of the paper is the "$\Sigma^*$-module completion," a
$\Sigma^*$-analogue of the selfdual completion of a $C^*$-module over a
$W^*$-algebra, which has an elegant uniqueness condition in the countably
generated case.
