Three-dimensional (3D) printing has expanded beyond the mere patterned deposition of melted solids, moving into areas requiring spatially structured soft matter—typically materials composed of polymers, colloids, surfactants, or living cells. The tunable and dynamically variable rheological properties of soft matter enable the high-resolution manufacture of soft structures. These rheological properties are leveraged in 3D printing techniques that employ sacrificial inks and sacrificial support materials, which go through reversible solid–fluid transitions under modest forces or other small perturbations. Thus, a sacrificial material can be used to shape a second material into a complex 3D structure, and then discarded. Here, we review the sacrificial materials and related methods used to print soft structures. We analyze data from the literature to establish manufacturing principles of soft matter printing, and we explore printing performance within the context of instabilities controlled by the rheology of soft matter materials.