Additive manufacturing (AM), widely known as 3D printing, is a method of manufacturing that forms parts from powder, wire or sheets in a process that proceeds layer by layer. Many techniques (using many different names) have been developed to accomplish this via melting or solid-state joining. In this review, these techniques for producing metal parts are explored, with a focus on the science of metal AM: processing defects, heat transfer, solidification, solid-state precipitation, mechanical properties and post-processing metallurgy. The various metal AM techniques are compared, with analysis of the strengths and limitations of each. Only a few alloys have been developed for commercial production, but recent efforts are presented as a path for the ongoing development of new materials for AM processes.