  Since its formulation by Hamaker et al., the radio interferometer measurement
equation (RIME) has provided a rigorous mathematical basis for the development
of novel calibration methods and techniques, including various approaches to
the problem of direction-dependent effects (DDEs). This series of papers aims
to place recent developments in the treatment of DDEs into one RIME-based
mathematical framework, and to demonstrate the ease with which the various
effects can be described and understood. It also aims to show the benefits of a
RIME-based approach to calibration.
  Paper I re-derives the RIME from first principles, extends the formalism to
the full-sky case, and incorporates DDEs. Paper II then uses the formalism to
describe self-calibration, both with a full RIME, and with the approximate
equations of older software packages, and shows how this is affected by DDEs.
It also gives an overview of real-life DDEs and proposed methods of dealing
with them. Applying this to WSRT data (Paper III) results in a noise-limited
image of the field around 3C 147 with a very high dynamic range (1.6 million),
and none of the off-axis artifacts that plague regular selfcal. The resulting
differential gain solutions contain significant information on DDEs, and can be
used for iterative improvements of sky models.
  Perhaps most importantly, sources as faint as 2 mJy have been shown to yield
meaningful differential gain solutions, and thus can be used as potential
calibration beacons in other DDE-related schemes.
