  In this paper we introduce a general version of the notion of Loewner chains
which comes from the new and unified treatment, given in [arXiv:0807.1594], of
the radial and chordal variant of the Loewner differential equation, which is
of special interest in geometric function theory as well as for various
developments it has given rise to, including the famous Schramm-Loewner
evolution. In this very general setting, we establish a deep correspondence
between these chains and the evolution families introduced in
[arXiv:0807.1594]. Among other things, we show that, up to a Riemann map, such
a correspondence is one-to-one. In a similar way as in the classical Loewner
theory, we also prove that these chains are solutions of a certain partial
differential equation which resembles (and includes as a very particular case)
the classical Loewner-Kufarev PDE.
