Most additive manufacturing processes today operate by printing voxels (3D pixels) serially point-by-point to build up a 3D part. In some more recently-developed techniques, for example optical printing methods such as projection stereolithography [Zheng et al. 2012], [Tumbleston et al. 2015], parts are printed layer-by-layer by curing full 2d (very thin in one dimension) layers of the 3d part in each print step. There does not yet exist a technique which is able to print arbitrarily-defined 3D geometries in a single print step. If such a technique existed, it could be used to expand the range of printable geometries in additive manufacturing and relax constraints on factors such as overhangs in topology optimization. It could also vastly increase print speed for 3D parts. In this work, we develop the principles for an approach for single exposure 3D printing of arbitrarily defined geometries. The approach, termed Computed Axial Lithgography (CAL), is based on tomographic reconstruction, with mathematical optimization to generate a set of projections to optically define an arbitrary dose distribution within a target volume. We demonstrate the potential ability of the technique to print 3D parts using a prototype CAL system based on sequential illumination from many angles. We also propose new hardware designs which will help us to realize true single-shot arbitrary-geometry 3D CAL.