  This chapter provides a principled introduction to uncertainty relations
underlying sparse signal recovery. We start with the seminal work by Donoho and
Stark, 1989, which defines uncertainty relations as upper bounds on the
operator norm of the band-limitation operator followed by the time-limitation
operator, generalize this theory to arbitrary pairs of operators, and then
develop -- out of this generalization -- the coherence-based uncertainty
relations due to Elad and Bruckstein, 2002, as well as uncertainty relations in
terms of concentration of $1$-norm or $2$-norm. The theory is completed with
the recently discovered set-theoretic uncertainty relations which lead to best
possible recovery thresholds in terms of a general measure of parsimony, namely
Minkowski dimension. We also elaborate on the remarkable connection between
uncertainty relations and the "large sieve", a family of inequalities developed
in analytic number theory. It is finally shown how uncertainty relations allow
to establish fundamental limits of practical signal recovery problems such as
inpainting, declipping, super-resolution, and denoising of signals corrupted by
impulse noise or narrowband interference. Detailed proofs are provided
throughout the chapter.
