Three-dimensional (3D) bioprinting is a promising technology to produce tissue-like structures, but a lack of diversity in bioinks is a major limitation. Ideally each cell type would be printed in its own customizable bioink. To fulfill this need for a universally applicable bioink strategy, we developed a versatile, bioorthogonal bioink crosslinking mechanism that is cell compatible and works with a range of polymers. We term this family of materials UNIversal, Orthogonal Network (UNION) bioinks. As demonstration of UNION bioink versatility, gelatin, hyaluronic acid (HA), recombinant elastin-like protein (ELP), and polyethylene glycol (PEG) were each used as backbone polymers to create inks with storage moduli spanning 200 to 10,000 Pa. Because UNION bioinks are crosslinked by a common chemistry, multiple materials can be printed together to form a unified, cohesive structure. This approach is compatible with any support bath that enables diffusion of UNION crosslinkers. Both matrix-adherent human corneal mesenchymal stromal cells and non-matrix-adherent human induced pluripotent stem cell-derived neural progenitor spheroids were printed with UNION bioinks. The cells retained high viability and expressed characteristic phenotypic markers after printing. Thus, UNION bioinks are a versatile strategy to expand the toolkit of customizable materials available for 3D bioprinting.