The range of applications for additive manufacturing is expanding quickly, including mass production of athletic footwear parts1, dental ceramics2 and aerospace components3 as well as fabrication of microfluidics4, medical devices5, and artificial organs6. The light-induced additive manufacturing techniques7 used are particularly successful owing to their high spatial and temporal control, but such techniques still share the common motifs of pointwise or layered generation, as do stereolithography8, laser powder bed fusion9, and continuous liquid interface production10 and its successors11,12. Volumetric 3D printing13-20 is the next step onward from sequential additive manufacturing methods. Here we introduce xolography, a dual colour technique using photoswitchable photoinitiators to induce local polymerization inside a confined monomer volume upon linear excitation by intersecting light beams of different wavelengths. We demonstrate this concept with a volumetric printer designed to generate three-dimensional objects with complex structural features as well as mechanical and optical functions. Compared to state-of-the-art volumetric printing methods, our technique has a resolution about ten times higher than computed axial lithography without feedback optimization, and a volume generation rate four to five orders of magnitude higher than two-photon photopolymerization. We expect this technology to transform rapid volumetric production for objects at the nanoscopic to macroscopic length scales.