Two limitations of additive manufacturing methods that arise from layer-based fabrication are slow speed and geometric constraints (which include poor surface quality). Both limitations are overcome in the work reported here, introducing a new volumetric additive fabrication paradigm that produces photopolymer structures with complex nonperiodic three-dimensional geometries on a time scale of seconds. We implement this approach using holographic patterning of light fields, demonstrate the fabrication of a variety of structures, and study the properties of the light patterns and photosensitive resins required for this fabrication approach. The results indicate that low-absorbing resins containing ~0.1% photoinitiator, illuminated at modest powers (~10 to 100 mW), may be successfully used to build full structures in ~1 to 10 s.