I will review a nonlocal gravity model proposed by Deser and Woodard to explain the current phase of cosmic acceleration without dark energy. I will present how the model reproduces the LambdaCDM expansion history without a cosmological constant and what is its prediction for cosmic structure formation, which turned out to be in better agreement with data than the one predicted by LambdaCDM. The studies of the structure formation were carried out by two groups, one in a localized formulation employed by Nersisyan, Fernandez Cid and Amendola and the other in the original nonlocal form by Dodelson and myself. By comparing these two formulations, I will show that they give the same solutions for perturbation variables provided the boundary conditions are set the same. I will also briefly discuss some conceptual issues in nonlocal gravity (including the Deser-Woodard model and other classes of nonlocal models introducing a new mass scale) such as causality, degrees of freedom and stability.