The continuous development of novel materials for biomedical applications is resulting in an increasingly better prognosis for patients. The application of more advanced materials relates to fewer complications and a desirable higher percentage of successful treatments. New, innovative materials being considered for biomedical applications are metallic alloys with an amorphous internal structure called metallic glasses. They are currently in a dynamic phase of development both in terms of formulating new chemical compositions and testing their properties in terms of intended biocompatibility. This review article intends to synthesize the latest research results in the field of biocompatible metallic glasses to create a more coherent picture of these materials. It summarizes and discusses the most recent findings in the areas of mechanical properties, corrosion resistance, in vitro cellular studies, antibacterial properties, and in vivo animal studies. Results are collected mainly for the most popular metallic glasses manufactured as thin films, coatings, and in bulk form. Considered materials include alloys based on zirconium and titanium, as well as new promising ones based on magnesium, tantalum, and palladium. From the properties of the examined metallic glasses, possible areas of application and further research directions to fill existing gaps are proposed.