Extrusion-based additive manufacturing (AM) technologies (i.e., fused filament fabrication, FFF) mostly produce thermoplastic parts. However, producing metallic or ceramic parts by FFF is also a sintered-based AM process. FFF for metallic parts can be divided into five steps: (1) raw materials selection and feedstock mixture (including palletization), (2) filament production (extrusion), (3) production of AM components using the filament extrusion process, (4) debinding, and (5) sintering. These steps are interrelated, where the parameters interact with the other ones and have a key role in the integrity and quality of the final metallic parts. FFF can produce high-accuracy and complex metallic parts, potentially revolutionizing the manufacturing industry and taking AM components to a new level. In the FFF technology for metallic materials, material compatibility, production quality, and cost-effectiveness are the challenges to overcome to make it more competitive compared to other AM technologies, like the laser processes. This review provides a comprehensive overview of the recent developments in FFF for metallic materials, including the metals and binders used, the challenges faced, potential applications, and the impact of FFF on the manufacturing (prototyping and end parts), design freedom, customization, sustainability, supply chain, among others.