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We report a theoretical study on the deviations of the Meissner penetration depth λ(T ) from its London value in d-wave superconductors at low temperatures. The difference arises from low-energy surface Andreev bound states. The temperature dependent penetration depth is shown to go through a minimum at the temperature Tm0 ∼ ξ0/λ0Tc if the broadening of the bound states is small. The minimum will straighten out when the broadening reaches Tm0. The impurity scattering sets up the low-temperature anomalies of the penetration depth and destroys them when the mean free path is not sufficiently large. A phase transition to a state with a spontaneous surface supercurrent is investigated and its critical temperature determined in the absence of a subdominant channel activated at low temperatures near the surface. Nonlinear corrections from Andreev low-energy bound states to the penetration length are obtained and shown, on account of their broadening, to be small in the Meissner state of strong type II superconductors.