Traditional optical imaging faces an unavoidable trade-off between resolution and depth of field (DOF). To increase resolution, high numerical apertures (NAs) are needed, but the associated large angular uncertainty results in a limited range of depths that can be put in sharp focus. Plenoptic imaging was introduced a few years ago to remedy this trade-off. To this aim, plenoptic imaging reconstructs the path of light rays from the lens to the sensor. However, the improvement offered by standard plenoptic imaging is practical and not fundamental: The increased DOF leads to a proportional reduction of the resolution well above the diffraction limit imposed by the lens NA. In this Letter, we demonstrate that correlation measurements enable pushing plenoptic imaging to its fundamental limits of both resolution and DOF. Namely, we demonstrate maintaining the imaging resolution at the diffraction limit while increasing the depth of field by a factor of 7. Our results represent the theoretical and experimental basis for the effective development of promising applications of plenoptic imaging.