Here it is demonstrated that nanofabricated structures can be used as electron optical elements in new types of electron interferometers. This enables novel investigations with electrons analogous to experiments in light and atom optics. Far field diffraction from a single nanograting is used to examine the force on a charge moving in close proximity to a surface. Near field diffraction from the nanograting is investigated in a Talbot interferometer. It is found that electron waves form replicas of the grating in free space, and these replicas can be de-magnified using illumination by a converging beam. An electron Lau interferometer has the same grating configuration as the Talbot interferometer, but uses spatially incoherent beams that give rise to drastically different interference behavior. A single optical theory is developed to efficiently model a variety of grating interferometers under a diverse set of illumination conditions, and it is used to understand the experiments described here. Applications for these new interferometers are discussed, as well as possible directions for future research.