The limited range of mechanical responses achievable by materials compatible with additive manufacturing hinders the 3D printing of continuum soft robots with programmed motion. This paper describes the rapid design and fabrication of low-density, 3D-architected soft machines (ASMs) by combining Voronoi tessellation and additive manufacturing. On tendon-based actuation, ASMs deform according to the topologically encoded buckling of their structure to produce a wide range of motions (contraction, twisting, bending, and cyclic motion). ASMs exhibiting densities as low as 0.094 g cm −3 (≈8% of bulk polymer) can be rapidly built by the stereolithographic 3D printing of flexible photopolymers or the injection molding of elastomers. The buckling of ASMs can be programmed by inducing gradients in the thickness of their flexible beams or by the localized enlargement of the Voronoi cells to generate complex motions such as multi-finger gripping or quadrupedal locomotion. The topological architecture of these low-density soft robots confers them with the stiffness necessary to recover their original shape even after ultrahigh compression (400%) and extension (500%). ASMs expand the range of mechanical properties currently achievable by 3D printed or molded materials to enable the fabrication of soft machines with auxetic mechanical metamaterial properties.