  Galaxy-galaxy lensing (GGL) measures the 2-point cross-correlation between
galaxies and mass in the Universe. In this work we seek to generalise this
effect by considering the third-order correlations between galaxies and mass:
galaxy-galaxy-galaxy lensing. Third-order correlations in the cosmic shear
field have recently been reported in the VIRMOS-DESCART and CTIO surveys. Such
data should also be ideal for measuring galaxy-galaxy-galaxy lensing. Indeed,
the effects of these higher-order correlations may have already been detected
in recent studies of galaxy-galaxy lensing. In this paper we lay out the basic
relations for third-order cross correlations and their projections and
introduce a new set of scale dependent third-order bias parameters. We define
three new observables: two galaxy-shear-shear correlation functions, $G_{\pm}$,
and a galaxy-galaxy-shear correlation, ${\cal G}$. We relate these to the
various projected cross-bispectra and give practical estimators for their
measurement. We note that the observational signature of these correlators is
simply the excess shear-shear correlation measured about foreground galaxies
(for $G_{\pm}$) and the average tangential shear around foreground galaxy pairs
(for $\cal G$). Finally we derive expressions for the third order aperture mass
statistics in terms of both the cross-bispectra and the real-space correlation
functions. (Abridged)
