  After a short exposition of the basic properties of the tautological ring of
the moduli space of genus g Deligne-Mumford stable curves with n markings, we
explain three methods of detecting non-tautological classes in cohomology. The
first is via curve counting over finite fields. The second is by obtaining
length bounds on the action of the symmetric group S_n on tautological classes.
The third is via classical boundary geometry. Several new non-tautological
classes are found.
