  The "expansion by regions" is a method of asymptotic expansion developed by
Beneke and Smirnov in 1997. It expands the integrand according to the scaling
prescriptions of a set of regions and integrates all expanded terms over the
whole integration domain. This method has been applied successfully to many
complicated loop integrals, but a general proof for its correctness has still
been missing. This paper shows how the expansion by regions manages to
reproduce the exact result correctly in an expanded form and clarifies the
conditions on the choice and completeness of the considered regions. A
generalized expression for the full result is presented that involves
additional overlap contributions. These extra pieces normally yield scaleless
integrals which are consistently set to zero, but they may be needed depending
on the choice of the regularization scheme. While the main proofs and formulae
are presented in a general and concise form, a large portion of the paper is
filled with simple, pedagogical one-loop examples which illustrate the
peculiarities of the expansion by regions, explain its application and show how
to evaluate contributions within this method.
