  From the simplest point of view, transseries are a new kind of expansion for
real-valued functions. But transseries constitute much more than that--they
have a very rich (algebraic, combinatorial, analytic) structure. The set of
transseries is a large ordered field, extending the real number field, and
endowed with additional operations such as exponential, logarithm, derivative,
integral, composition. Over the course of the last 20 years or so, transseries
have emerged in several areas of mathematics: asymptotic analysis, model
theory, computer algebra, surreal numbers. This paper is an exposition for the
non-specialist mathematician.
  All a mathematician needs to know in order to apply transseries.

Rings and Algebras;Classical Analysis and ODEs;History and Overview