  We explore spectral duality in the context of measures in $\br^n$, starting
with partial differential operators and Fuglede's question (1974) about the
relationship between orthogonal bases of complex exponentials in $L^2(\Omega)$
and tiling properties of $\Omega$, then continuing with affine iterated
function systems. We review results in the literature from 1974 up to the
present, and we relate them to a general framework for spectral duality for
pairs of Borel measures in $\br^n$, formulated first by Jorgensen and Pedersen.
