The production rate and kinematics of photons produced in association with Z bosons are studied using 2fb-1 of pp̄ collision data collected at the Collider Detector at Fermilab. The cross section for pp̄→"+-γ+X (where the leptons " are either muons or electrons with dilepton mass M>40GeV/c2, and where the photon has transverse energy ETγ>7GeV and is well separated from the leptons) is 4.6±0.2(stat)±0.3(syst)±0.3(lum)pb, which is consistent with standard model expectations. We use the photon ET distribution from Zγ events where the Z has decayed to μ+μ -, e+e-, or νν̄ to set limits on anomalous (non standard model) trilinear couplings between photons and Z bosons. © 2010 The American Physical Society.