  Several classes of tempered measures are characterised that are eigenmeasures
of the Fourier transform, the latter viewed as a linear operator on (generally
unbounded) Radon measures on $\RR^d$. In particular, we classify all periodic
eigenmeasures on $\RR$, which gives an interesting connection with the discrete
Fourier transform and its eigenvectors, as well as all eigenmeasures on $\RR$
with uniformly discrete support. An interesting subclass of the latter emerges
from the classic cut and project method for aperiodic Meyer sets. Finally, we
construct a large class of eigenmeasures with locally finite support that is
not uniformly discrete and has large gaps around $0$.
