  We extend ideas developed for the loop representation of quantum gravity to
diffeomorphism-invariant gauge theories coupled to fermions. Let P -> Sigma be
a principal G-bundle over space and let F be a vector bundle associated to P
whose fiber is a sum of continuous unitary irreducible representations of the
compact connected gauge group G, each representation appearing together with
its dual. We consider theories whose classical configuration space is A x F,
where A is the space of connections on P and F is the space of sections of F,
regarded as a collection of Grassmann-valued fermionic fields. We construct the
`quantum configuration space a x f as a completion of A x F. Using this we
construct a Hilbert space L^2(a x f) for the quantum theory on which all
automorphisms of P act as unitary operators, and determine an explicit `spin
network basis' of the subspace L^2((a x f)/G) consisting of gauge-invariant
states. We represent observables constructed from holonomies of the connection
along paths together with fermionic fields and their conjugate momenta as
operators on L^2((a x f)/G). We also construct a Hilbert space H_diff of
diffeomorphism-invariant states using the group averaging procedure of
Ashtekar, Lewandowski, Marolf, Mourao and Thiemann.
