  We present new techniques for finding anomaly-free sets of fermions. Although
the anomaly cancellation conditions typically include cubic equations with
integer variables that cannot be solved in general, we prove by construction
that any chiral set of fermions can be embedded in a larger set of fermions
which is chiral and anomaly-free. Applying these techniques to extensions of
the Standard Model, we find anomaly-free models that have arbitrary quark and
lepton charges under an additional U(1) gauge group.
