Even though a fundamental Lagrangian is local, quantum loops of massless particles give rise to nonlocal effective field equations. Inspired by this fact, several classes of nonlocal gravity models have been proposed in an attempt to describe observed cosmic motions without invoking the hypothetical substances, dark energy and/or dark matter. First, I will review two nonlocal models replacing dark energy, one proposed by Deser and Woodard and the other by Maggiore and Mancarella. I will present how they generate the current cosmic acceleration without dark energy and how the structure formations predicted by these two differ from each other. Second, I will review one nonlocal model replacing dark matter, proposed by Deffayet, Esposito-Farese and Woodard. For this case, I will focus on how it reproduces the Tully-Fisher relation and sufficient weak lensing. Along the way, I will also discuss some issues concerning nonlocal gravity and a possible connection between the nonlocality and infrared quantum loop corrections.