Mechanical metamaterials can be defined as a class of architected materials that exhibit unprecedented mechanical properties derived from designed artificial architectures rather than their constituent materials. While macroscale and simple layouts can be realized by conventional top-down manufacturing approaches, many of the sophisticated designs at various length scales remain elusive, due to the lack of adequate manufacturing methods. Recent progress in additive manufacturing (AM) has led to the realization of a myriad of novel metamaterial concepts. AM methods capable of fabricating microscale architectures with high resolution, arbitrary complexity, and high feature fidelity have enabled the rapid development of architected metamaterials and drastically reduced the design-computation and experimental-validation cycle. This paper first provides a detailed review of various topologies based on the desired mechanical properties, including stiff, strong, and auxetic (negative Poisson's ratio) metamaterials, followed by a discussion of the AM technologies capable of fabricating these metamaterials. Finally, we discuss current challenges and recommend future directions for AM and mechanical metamaterials.