  We develop a new, more functorial construction for the basic theory of limit
linear series, which provides a compactification of the Eisenbud-Harris theory,
and shows promise for generalization to higher-dimensional varieties and
higher-rank vector bundles. We also give a result on lifting linear series from
characteristic p to characteristic 0. In an appendix, in order to obtain the
necessary dimensional lower bounds on our limit linear series scheme we develop
a theory of "linked Grassmannians;" these are schemes parametrizing sub-bundles
of a sequence of vector bundles which map into one another under fixed maps of
the ambient bundles.
