Stainless steel, cobalt-chromium alloy, and titanium alloys have recently been found to be extensively used in medical applications. Additive manufacturing (AM) techniques have played a significant role in creating complex three-dimensional (3D) shapes for medical parts and/or devices like implants, prostheses, and other instruments using these metal alloys. Metal implants can be created utilizing a variety of AM processes and a range of metal shapes, starting with powder, wire, and sheet. Medical applications require biocompatibility, which is dependent on many criteria such as chemical composition, surface characteristics, porosity, and microstructure. Post-processing techniques such as surface modifications, coatings, functionalization with bioactive molecules, and surface texturing and patterning are considered to improve the biocompatibility of metal implants. In vitro and in vivo methods are used to assess the biocompatibility of these implants. This research also discusses case studies and clinical applications, highlighting the sustainability of their development. However, challenges and limitations were identified during the AM process and post-production, along with potential risks. This study concludes that the mentioned metal alloys demonstrate excellent mechanical and functional properties and optimal biocompatibility for biomedical applications using metal AM techniques. According to the study's findings, metal additive manufacturing (MAM) techniques can produce metal implants with good mechanical and functional qualities and optimal biocompatibility.