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In recent years, the increasing complexity and safety-critical nature of robotic tasks have highlighted the importance of accurate and reliable robot models. This trend has led to a growing belief that, given enough data, traditional physics-based robot models can be replaced by appropriately trained deep networks or their variants. Simultaneously, there has been a renewed interest in physics-based simulation, fueled by the widespread use of simulators to train reinforcement learning algorithms in the sim-to-real paradigm. The primary objective of this review is to present a unified perspective on the process of determining robot models from data, commonly known as system identification or model learning in different subfields. The review aims to illuminate the key challenges encountered and highlight recent advancements in system identification for robotics. Specifically, we focus on recent breakthroughs that leverage the geometry of the identification problem and incorporate physics-based knowledge beyond mere first-principles model parameterizations. Through these efforts, we strive to provide a contemporary outlook on this problem, bridging classical findings with the latest progress in the field.