  In this paper we introduce a general version of the Loewner differential
equation which allows us to present a new and unified treatment of both the
radial equation introduced in 1923 by K. Loewner and the chordal equation
introduced in 2000 by O. Schramm. In particular, we prove that evolution
families in the unit disc are in one to one correspondence with solutions to
this new type of Loewner equations. Also, we give a Berkson-Porta type formula
for non-autonomous weak holomorphic vector fields which generate such Loewner
differential equations and study in detail geometric and dynamical properties
of evolution families.
