Bioprinting is rapidly being adopted as a major method for fabricating tissue engineering constructs. Through the precise deposition of cell- and bioactive molecule-laden materials, bioprinting offers researchers a means to create biological constructs with enhanced spatial complexity that more closely mimics native tissue. The vast majority of materials used in bioprinting have been polymers due to their suitability toward resembling the cellular environment and the variety of methods available to process polymeric systems in ambient or relatively mild chemical and environmental conditions. In this review, we will discuss in detail the wide variety of natural and synthetic polymers that have been employed as inks in bioprinting. We will review recent bioprinting innovations, such as increasing architectural complexity and cell viability in heterogeneous tissue constructs, which allow for the investigation of biological questions that could not be addressed before. We will also survey nascent fields of study that promise to further advance the development of novel biofabrication technologies in the field, such as 4D bioprinting and the inclusion of nanomaterials. To conclude, we will examine some of the necessary steps that must take place to bring this technology to commercial markets and facilitate its use in clinical therapies.