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As hardware capabilities have become sufficient for many applications, and as computing has become more powerful, algorithms that support fluent and expressive use of pHRI systems have begun to play a prominent role in determining the systems\u2019 usefulness. In this review, we describe a selection of representative algorithmic approaches that regulate and interpret pHRI, describing the progression from algorithms based on physical analogies, such as admittance control, to computational methods based on higher-level reasoning, which take advantage of multimodal communication channels. Existing algorithmic approaches largely enable task-specific pHRI, but they do not generalize to versatile human\u2013robot collaboration. 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