  In this work it is introduced the notion of regular Fredholm pair, i.e. a
Fredholm pair whose operators are regular. The main properties of these objects
are studied, and what is more, they are entirely classified. Furthermore, the
index of a Fredholm pair turns out to be an extremely useful tool in the
description of the aforementioned objects. Finally, regular Fredholm pairs are
characterized in terms of regular Fredholm symmetrical pairs, exact chains of
multiplication operators, and invertible Banach space operators.
