  Translation surfaces can be defined in an elementary way via polygons, and
arise naturally in in the study of various basic dynamical systems. They can
also be defined as Abelian differentials on Riemann surfaces, and have moduli
spaces called strata that are related to the moduli space of Riemann surfaces.
There is a GL(2,R) action on each stratum, and to solve most problems about a
translation surface one must first know the closure of its orbit under this
action. Furthermore, these orbit closures are of fundamental interest in their
own right, and are now known to be algebraic varieties that parameterize
translation surfaces with extraordinary algebro-geometric and flat properties.
The study of orbit closures has greatly accelerated in recent years, with an
influx of new tools and ideas coming diverse areas of mathematics.
  This survey is an invitation for mathematicians from different backgrounds to
become familiar with the subject. Little background knowledge, beyond the
definition of a Riemann surface and its cotangent bundle, is assumed, and top
priority is given to presenting a view of the subject that is at once
accessible and connected to many areas of mathematics.
