  We present a comprehensive and updated comparison with cosmological
observations of two non-local modifications of gravity previously introduced by
our group, the so called RR and RT models. We implement the background
evolution and the cosmological perturbations of the models in a modified
Boltzmann code, using CLASS. We then test the non-local models against the {\em
Planck} 2015 TT, TE, EE and Cosmic Microwave Background (CMB) lensing data,
isotropic and anisotropic Baryonic Acoustic Oscillations (BAO) data, JLA
supernovae, $H_0$ measurements and growth rate data, and we perform Bayesian
parameter estimation. We then compare the RR, RT and $\Lambda$CDM models, using
the Savage-Dickey method. We find that the RT model and $\Lambda$CDM perform
equally well, while the performance of the RR model with respect to
$\Lambda$CDM depends on whether or not we include a prior on $H_0$ based on
local measurements.
