  We propose a generalization of Artin's definition of algebraic stack, which
we call {\em geometric $n$-stack}. The main observation is that there is an
inductive structure to the definition whereby the ingredients for the
definition of geometric $n$-stack involve only $n-1$-stacks and so are already
previously defined. We use this inductive structure to obtain some basic
properties. We look at maps from a projective variety into certain such
$n$-stacks, and obtain an interpretation of the Brill-Noether locus as the set
of points of a geometric $n$-stack. At the end we explain how this provides a
context for looking at de Rham theory for higher nonabelian cohomology, how one
can define the Hodge filtration and so on.
