  In holomorphic dynamics, complex box mappings arise as first return maps to
well-chosen domains. They are a generalization of polynomial-like mapping,
where the domain of the return map can have infinitely many components. They
turned out to be extremely useful in tackling diverse problems. The purpose of
this paper is:
  -To illustrate some pathologies that can occur when a complex box mapping is
not induced by a globally defined map and when its domain has infinitely many
components, and to give conditions to avoid these issues.
  -To show that once one has a box mapping for a rational map, these conditions
can be assumed to hold in a very natural setting. Thus we call such complex box
mappings dynamically natural.
  -Many results in holomorphic dynamics rely on an interplay between
combinatorial and analytic techniques: (*)the Enhanced Nest by
Kozlovski-Shen-van Strien; (*)the Covering Lemma by Kahn-Lyubich; (*)the
QC-Criterion, the Spreading Principle. The purpose of this paper is to make
these tools more accessible so that they can be used as a 'black box', so one
does not have to redo the proofs in new settings.
  -To give an intuitive, but also rather detailed, outline of the proof of the
following results by Kozlovski-van Strien for non-renormalizable dynamically
natural box mappings: (*)puzzle pieces shrink to points; (*)topologically
conjugate non-renormalizable polynomials and box mappings are quasiconformally
conjugate.
  -We prove the fundamental ergodic properties for dynamically natural box
mappings. This leads to some necessary conditions for when such a box mapping
supports a measurable invariant line field on its filled Julia set. These
mappings are the analogues of Lattes maps in this setting.
  -We prove a version of Mane's Theorem for complex box mappings concerning
expansion along orbits of points that avoid a neighborhood of the set of
critical points.
