Ultrathin dielectric gaps between metals can trap plasmonic optical modes with surprisingly low loss and with volumes below 1 nm3. We review the origin and subtle properties of these modes, and show how they can be well accounted for by simple models. Particularly important is the mixing between radiating antennas and confined nanogap modes, which is extremely sensitive to precise nanogeometry, right down to the single-atom level. Coupling nanogap plasmons to electronic and vibronic transitions yields a host of phenomena including single-molecule strong coupling and molecular optomechanics, opening access to atomic-scale chemistry and materials science, as well as quantum metamaterials. Ultimate low-energy devices such as robust bottom-up assembled single-atom switches are thus in prospect.