The most prevalent form of bioprinting—extrusion bioprinting—can generate structures from a diverse range of materials and viscosities. It can create personalized tissues that aid in drug testing and cancer research when used in combination with natural bioinks. This paper reviews natural bioinks and their properties and functions in hard and soft tissue engineering applications. It discusses agarose, alginate, cellulose, chitosan, collagen, decellularized extracellular matrix, dextran, fibrin, gelatin, gellan gum, hyaluronic acid, Matrigel, and silk. Multi-component bioinks are considered as a way to address the shortfalls of individual biomaterials. The mechanical, rheological, and cross-linking properties along with the cytocompatibility, cell viability, and printability of the bioinks are detailed as well. Future avenues for research into natural bioinks are then presented.