Three-dimensional (3D) printing is becoming an increasingly common technique to fabricate scaffolds and devices for tissue engineering applications. This is due to the potential of 3D printing to provide patient-specific designs, high structural complexity, rapid on-demand fabrication at a low-cost. One of the major bottlenecks that limits the widespread acceptance of 3D printing in biomanufacturing is the lack of diversity in "biomaterial inks". Printability of a biomaterial is determined by the printing technique. Although a wide range of biomaterial inks including polymers, ceramics, hydrogels and composites have been developed, the field is still struggling with processing of these materials into self-supporting devices with tunable mechanics, degradation, and bioactivity. This review aims to highlight the past and recent advances in biomaterial ink development and design considerations moving forward. A brief overview of 3D printing technologies focusing on ink design parameters is also included.