  Hodge correlators are complex numbers given by certain integrals assigned to
a smooth complex curve. We show that they are correlators of a Feynman
integral, and describe the real mixed Hodge structure on the pronilpotent
completion of the fundamental group of the curve. We introduce motivic
correlators, which are elements of the motivic Lie algebra and whose periods
are the Hodge correlators. They describe the motivic fundamental group of the
curve. We describe variations of real mixed Hodge structures on a variety by
certain connections on the product of the variety by an afine line. We call
them twistor connections. Generalising this, we suggest a DG enhancement of the
subcategory of Saito's Hodge complexes with smooth cohomology. We show that
when the curve varies, the Hodge correlators are the coefficients of the
twistor connection describing the corresponding variation of real MHS. Examples
of the Hodge correlators include classical and elliptic polylogarithms, and
their generalizations. The simplest Hodge correlators on the modular curves are
the Rankin-Selberg integrals. Examples of the motivic correlators include
Beilinson's elements in the motivic cohomology, e.g. the ones delivering the
Beilinson - Kato Euler system on modular curves.
