  In complex systems, it is often the case that the region of interest forms
only one part of a much larger system. The idea of joining two different
quantum simulations - a high level calculation on the active region of
interest, and a low level calculation on its environment - formally defines a
quantum embedding. While any combination of techniques constitutes an
embedding, several rigorous formalisms have emerged that provide for exact
feedback between the embedded system and its environment. These three
formulations: it density functional embedding, Green's function embedding, and
density matrix embedding, respectively use the single-particle density,
single-particle Green's function, and single-particle density matrix as the
quantum variables of interest. Many excellent reviews exist covering these
methods individually. However, a unified presentation of the different
formalisms is so far lacking. Indeed, the various languages commonly used:
functional equations for density functional embedding; diagrammatics for
Green's function embedding; and entanglement arguments for density matrix
embedding, make the three formulations appear vastly different. In this
account, we introduce the basic equations of all three formulations in such a
way as to highlight their many common intellectual strands. While we focus
primarily on a straightforward theoretical perspective, we also give a brief
overview of recent applications, and possible future developments.
