Surgical limitations require alternative methods of repairing and replacing diseased and damaged tissue. Regenerative medicine is a growing area of research with engineered tissues already being used successfully in patients. However, the demand for such tissues greatly outweighs the supply and a fast and accurate method of production is still required. 3D bioprinting offers precision control as well as the ability to incorporate biological cues and cells directly into the material as it is being fabricated. Having precise control over scaffold morphology and chemistry is a significant step towards controlling cellular behaviour, particularly where undifferentiated cells, i.e., stem cells, are used. This level of control in the early stages of tissue development is crucial in building more complex systems that morphologically and functionally mimic in vivo tissue. Here we review 3D printing hydrogel materials for tissue engineering purposes and the incorporation of cells within them. Hydrogels are ideal materials for cell culture. They are structurally similar to native extracellular matrix, have a high nutrient retention capacity, allow cells to migrate and can be formed under mild conditions. The techniques used to produce these materials, as well as their benefits and limitations, are outlined.