Recent advances in three-dimensional (3D) printing technologies have led to a rapid expansion of applications from the creation of anatomical training models for complex surgical procedures to the printing of tissue engineering constructs. In addition to achieving the macroscale geometry of organs and tissues, a print layer thickness as small as 20 μm allows for reproduction of the microarchitectures of bone and other tissues. Techniques with even higher precision are currently being investigated to enable reproduction of smaller tissue features such as hepatic lobules. Current research in tissue engineering focuses on the development of compatible methods (printers) and materials (bioinks) that are capable of producing biomimetic scaffolds. In this review, an overview of current 3D printing techniques used in tissue engineering is provided with an emphasis on the printing mechanism and the resultant scaffold characteristics. Current practical challenges and technical limitations are emphasized and future trends of bioprinting are discussed.