  The combinatorial theory of species developed by Joyal provides a foundation
for enumerative combinatorics of objects constructed from finite sets. In this
paper we develop an analogous theory for the enumerative combinatorics of
objects constructed from vector spaces over finite fields. Examples of these
objects include subspaces, flags of subspaces, direct sum decompositions, and
linear maps or matrices of various types. The unifying concept is that of a
q-species, defined to be a functor from the category of finite dimensional
vector spaces over a finite field with to the category of finite sets.
