In this paper we introduce a general version of the notion of
Loewner chains which comes from the new and unified treatment, given
in [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution
families and the Loewner equation I: the unit disk. To appear in J. Reine
Angew. Math.] 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 [Bracci, F., Contreras, M.D. and
Díaz-Madrigal, S.: Evolution families and the Loewner equation I:
the unit disk. To appear in J. Reine Angew. Math.]. 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.