The integrability problem consists in finding the class of functions a first
integral of a given planar polynomial differential system must belong to. We
recall the characterization of systems which admit an elementary or Liouvillian
first integral. We define {\it Weierstrass integrability} and we determine
which Weierstrass integrable systems are Liouvillian integrable. Inside this
new class of integrable systems there are non--Liouvillian integrable systems.