  In differential topology two smooth submanifolds $S_1$ and $S_2$ of euclidean
space are said to be transverse if the tangent spaces at each common point
together form a spanning set. The purpose of this article is to explore a much
more general notion of transversality pertaining to a collection of
submanifolds of euclidean space. In particular, we show that three seemingly
different concepts of transversality arising naturally in harmonic analysis,
are in fact equivalent. This result is an amalgamation of several recent works
on variants of the Brascamp--Lieb inequality, and we take the opportunity here
to briefly survey this growing area. This is not intended to be an exhaustive
account, and the choices made reflect the particular perspectives of the
authors.
