  In this paper, we show that Severi varieties parameterizing irreducible
reduced planar curves of a given degree and geometric genus are either empty or
irreducible in any characteristic. Following Severi's original idea, this gives
a new proof of the irreducibility of the moduli space of smooth projective
curves of a given genus in positive characteristic. It is the first proof that
involves no reduction to the characteristic zero case. As a further
consequence, we generalize Zariski's theorem to positive characteristic and
show that a general reduced planar curve of a given geometric genus is nodal.
