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We study the cosmology of Galileon modified gravity models in the linear perturbation regime. We derive the fully covariant and gauge invariant perturbed field equations using two different methods, which give consistent results, and solve them using a modified version of the CAMB code. We find that, in addition to modifying the background expansion history and therefore shifting the positions of the acoustic peaks in the cosmic microwave background power spectrum, the Galileon field can cluster strongly from early times, and causes the Weyl gravitational potential to grow, rather than decay, at late times. This leaves clear signatures in the low-/ cosmic microwave background power spectrum through the modified integrated Sachs-Wolfe effect, strongly enhances the linear growth of matter density perturbations and makes distinctive predictions for other cosmological signals such as weak lensing and the power spectrum of density fluctuations. The quasistatic approximation is shown to work quite well from small to the near-horizon scales. We demonstrate that Galileon models display a rich phenomenology due to the large parameter space and the sensitive dependence of the model predictions on the Galileon parameters. Our results show that some Galileon models are already ruled out by present data and that future higher significance galaxy clustering, integrated Sachs-Wolfe, and lensing measurements will place strong constraints on Galileon gravity. [ABSTRACT FROM AUTHOR]