Three-dimensional (3D) bioprinting has become a fast-developing research field in the last few years. Many different technical solutions are available, with extrusion-based printing being the most promising and versatile method. In addition, a variety of biomaterials are already available for 3D printing of live cells. The real challenge, however, remains bioprinting of macroscopic, volumetric constructs of well-defined structures since hydrogels used for cell-embedding must consist of rather soft materials. This article describes recent developments to overcome these limitations that prevent clinical applications of bioprinted human tissues. New approaches include technical solutions such as in situ cross-linking or gelation processes that now can be performed during the bioprinting process, modified bioinks that combine suitable viscosity and cytocompatible gelation mechanisms, and utilization of additional materials to provide mechanical strength to the cell-laden constructs.