  We study the spectrum of BPS particles on the Coulomb branch of
five-dimensional superconformal field theories (5d SCFTs) compactified on a
circle. By engineering these theories in M-theory on ${\mathbf X} \times S^1 $,
for ${\mathbf X}$ an isolated Calabi-Yau threefold singularity, we naturally
identify the BPS category of the 5d theory on a circle with the derived
category of coherent sheaves on a resolution of ${\mathbf X}$. It follows that
the BPS spectrum can be studied in terms of 5d BPS quivers, which are the
fractional-brane quivers for the singularity ${\mathbf X}$. 5d BPS quivers
generalize the well-studied 4d BPS quivers for 4d $\mathcal{N}{=}2$ gauge
theories that can be obtained from ${\mathbf X}$ in so-called geometric
engineering limits. We study the interplay between 4d and 5d BPS quivers in
detail. We particularly focus on examples when ${\mathbf X}$ is a toric
singularity, in which case the 5d BPS quiver is given in terms of a brane
tiling. For instance, the well-studied $Y^{p,q}$ brane tiling gives a 5d BPS
quiver for the $SU(p)_q$ 5d gauge theory. We present a conjecture about the
structure of the BPS spectra of a wide class of models, which we test in the
simple case of the 5d $SU(2)_0$ theory (more precisely, the $E_1$ SCFT). We
also argue that 5d UV dualities can be realized in terms of mutation sequences
on the BPS quivers, which are in turn interpreted as autoequivalences of the
BPS category.
