  [Abridged] This review article presents the progress made over the last
decade, since the introduction of effective field theories (EFTs) into
post-Newtonian (PN) gravity. These have been put forward in the context of
gravitational waves (GWs) from the compact binary inspiral. The mature
development of this interdisciplinary field has resulted in significant
advances of wide interest to physics at several levels serving various
purposes. The field has firmly demonstrated, that seemingly disparate physical
domains, such as quantum field theory and classical gravity, are related, and
that the EFT framework is a universal one, where it has been proven to supply a
robust methodology to boost progress in the development of PN theory. The
review is aimed at a broad audience, from general readers new to the field, to
specialists and experts in related subjects.
  The review begins with an overview of the introduction of EFTs into classical
gravity and their development. Then, the basic ideas, which form the conceptual
foundation of EFTs, are provided, and the strategy of a multi-stage EFT
framework, which is utilized for the PN binary inspiral problem, is outlined.
The main body of the review is then dedicated to presenting in detail the study
of each of the effective theories at each of the intermediate scales in the
problem, up to the actual GW observables. The review is concluded with the
multiple prospects of building on the progress in the field, and using further
modern field theory insights and tools, to specifically address the study of
GWs, as well as to broadly expand our fundamental understanding of gauge and
gravity theories across the classical and quantum regimes.
