Additive manufacturing (AM) is a layerwise production process that creates three-dimensional objects according to a digital model. This technology has demonstrated to be a promising alternative to conventional manufacturing methods for various industrial sectors, such as aerospace, automotive, biomedical, and energy. AM offers several advantages, like design flexibility, material efficiency, functional integration, and rapid prototyping. As regards metal parts, conventional AM techniques using infrared laser sources face some limitations in processing high-reflectivity and high-conductivity materials or alloys, such as aluminum, copper, gold, and silver. These materials have low absorption of infrared radiation, which results in unstable and shallow melt pools, poor surface quality, and high porosity. To overcome these challenges, green and blue laser sources have been proposed for AM processes. This review provides an overview of the recent developments and applications of green and blue laser sources for powder bed fusion of copper and its alloys, focusing on the effects of process parameters on the melt pool dynamics, microstructure formation, and thermal and electrical properties of the fabricated parts. This review also presents the main applications of AM of copper and its alloys together with potential opportunities for future developments.