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 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 ΛCDM models, using the Savage-Dickey method. We find that the RT model and ΛCDM perform equally well, while the performance of the RR model with respect to ΛCDM depends on whether or not we include a prior on H 0 based on local measurements. arXiv:1602.03558v2 [astro-ph.CO] 22 May 2016 2 This is analogous to the fact that, in a Deser-Woodard type model, a term of the form R µν f (2 −1 Rµν ) shows instabilities [60].3 See also app. B below, where we will discuss a related issue on the comparison with Lunar Laser Ranging, raised in [64].4 It should also be appreciated that a non-local model such as the RT or the RR model only introduces one new parameter m, which replaces the cosmological constant in ΛCDM. By comparison, bigravity replaces the cosmological constant by a set of 5 parameters βn, n = 0, . . . , 4 and also introduces a new Planck mass associated to the second metric, and viable solutions are searched tuning this parameter space. Similarly, in the Deser-Woodard model one tunes a whole function f (X).