  We study the properties of reflectionless measures for a Calder\'{o}n-Zygmund
operator T. Roughly speaking, these are measures $\mu$ for which T(\mu)
vanishes (in a weak sense) on the support of the measure. We describe the
relationship between certain well-known problems in harmonic analysis and
geometric measure theory and the classification of reflectionless measures. As
an application of our theory, we give a new proof of a recent theorem of
Eiderman, Nazarov, and Volberg, which states that in $\mathbb{R}^d$, the
s-dimensional Riesz transform of a non-trivial $s$-dimensional measure is
unbounded if $s\in (d-1,d)$.
