  We show how a number of well-known uncertainty principles for the Fourier
transform, such as the Heisenberg uncertainty principle, the Donoho--Stark
uncertainty principle, and Meshulam's non-abelian uncertainty principle, have
little to do with the structure of the Fourier transform itself. Rather, all of
these results follow from very weak properties of the Fourier transform (shared
by numerous linear operators), namely that it is bounded as an operator $L^1
\to L^\infty$, and that it is unitary. Using a single, simple proof template,
and only these (or weaker) properties, we obtain some new proofs and many
generalizations of these basic uncertainty principles, to new operators and to
new settings, in a completely unified way. Together with our general overview,
this paper can also serve as a survey of the many facets of the phenomena known
as uncertainty principles.
