Since their discovery in the 1960s, metallic glasses have continuously attracted much interest across the physics and materials science communities. In the forefront are their unique properties, which hold the alluring promise of broad application in fields as diverse as medicine, environmental science and engineering. However, a major obstacle to their wide-spread commercial use is their inherent temporal instability arising from underlying relaxation processes that can dramatically alter their physical properties. The result is a physical aging process which can bring about degradation of mechanical properties, namely through embrittlement and catastrophic mechanical failure. Understanding and controlling the effects of aging will play a decisive role in our on-going endeavor to advance the use of metallic glasses as structural materials, as well as in the more general comprehension of out-of-equilibrium dynamics in complex systems. This review presents an overview of the current state of the art in the experimental advances probing physical aging and relaxation processes in metallic glasses. Similarities and differences between other hard and soft matter glasses are highlighted. The topic is discussed in a multiscale approach, first presenting the key features obtained in macroscopic studies, then connecting them to recent novel microscopic investigations. Particular emphasis is put on the occurrence of distinct relaxation processes beyond the main structural process in viscous metallic melts and their fate upon entering the glassy state, trying to disentangle results and formalisms employed by the different groups of the glass-science community. A microscopic viewpoint is presented, in which physical aging manifests itself in irreversible atomic-scale processes such as avalanches and intermittent dynamics, ascribed to the existence of a plethora of metastable glassy states across a complex energy landscape. Future experimental challenges and the comparison with recent theoretical and numerical simulations are discussed as well.