"Our proposal addresses theoretical and phenomenological properties of large distance (“Infra-Red”, IR in the following) modifications of the gravitational interaction. Such modifications are motivated by two main reasons: firstly, to find alternative explanations to the presence of dark matter or dark energy in cosmology; secondly, to better understand the currently well accepted cosmological model, disentangling there what does from what does not depend on the large distance dynamics of gravity and extracting as much as possible new information on gravity from the latest cosmological observations. For the second goal, it matters to have at hand alternatives to the standard cosmological model based on general relativity, to serve as benchmarks. Very recently, new ideas have been proposed to modified gravity in the IR. First, a large class of scalar-tensor theories featuring the “k-mouflaging” of the scalar has been proposed and partly classified. Second, new kinds of massive gravities which might be devoid of the standard pathologies of those models have been discovered. Third, models of non local gravity have been proposed with many interesting features. In this proposal, we intend to better understand those constructions, in which the works of the applicant played a major role, and whose properties are largely unexplored. As transversal goals, we also intend to propose new ways to modify gravity in the IR, as well as to develop schemes to tests IR modifications of gravity against cosmological and gravitational data. The project will be lead by the applicant, four postdocs and two students."