One of the areas that have benefited the most from the advent of additive manufacturing is the development of customized cellular materials, scaffolds and lattices. Although these different groups of materials are typically considered separately, they can be categorized as mechanical metamaterials. Among the different additive manufacturing techniques, perhaps the most popular is that of Fused Filament Fabrication. Numerous works have been reported in the literature in which this fabrication technique has been used to produce such materials. Inspired by the increasing volume of work dealing with the subject, we present a review of the manufacturing and characterization of cellular and lattice-based mechanical metamaterials using Fused Filament Fabrication. An overview of the topologies, their effective mechanical properties and intrinsic manufacturing aspects are presented. The methods for failure analysis at different scales are also discussed. Finally, studies comparing the production of mechanical metamaterials using Fused Filament Fabrication and other additive manufacturing techniques are presented, in addition to recommendations and current trends in the production of these structures by Fused Filament Fabrication.