  As the sensitivity of current and future gravitational-wave detectors
improves, it will become possible to measure the evolution of the binary black
hole merger rate with redshift. Here, we combine detailed fits to
state-of-the-art dynamical models of binary black hole formation in dense star
clusters with a cosmological model of cluster formation across cosmic time. We
find a typical merger rate of 14 $\rm{Gpc}^{-3} \rm{yr}^{-1}$ in the local
universe, with a reasonable range of 4-18 $\rm{Gpc}^{-3} \rm{yr}^{-1}$,
depending on the rate of cluster disruption and the cluster initial mass
function. This rate increases by a factor of 6 to redshift $z=2.7$ before
declining at higher redshifts. We compare the merger rate from binaries
produced in clusters to similar estimates from isolated binaries and triples in
galactic fields, and discuss various ways that these different formation
channels could add up to the current merger rate observed by LIGO/Virgo.
