This review evaluates hydrogel-forming polymers that are suitable for soft tissue engineering with a focus on materials that can be fabricated using additive manufacturing (3D-printing). An overview of the specific material requirements for hydrogel-based tissue engineering constructs is presented. This is followed by an explanation of the various hydrogel-forming polymer classes that includes a detailed examination of material properties that are critical for extrusion printing. Specifically, mechanisms for hydrogel formation, degradation, and biological response, activity and compatibility are explored. A discussion of extrusion printing strategies for printable hydrogel-forming polymers is then presented in conjunction with a list of considerations to guide future tissue engineering developments.