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Spin-orbit coupling (SOC) relates to the interaction between an electron's motion and its spin, and is ubiquitous in solid-state systems. Although the effect of SOC in normal-state phenomena has been extensively studied, its role in superconducting hybrid structures and devices opens many unexplored questions. In conjunction with broken symmetries and material inhomogeneities within superconducting hybrid structures, SOC may have additional contributions, beyond its effects in homogenous materials. Remarkably, even with well-established magnetic or nonmagnetic materials and conventional s-wave spin-singlet superconductors, SOC leads to emergent phenomena including equal-spin triplet pairing and topological superconductivity (hosting Majorana states), a modified current-phase relationship in Josephson junctions, and nonreciprocal transport. SOC is also responsible for transforming quasiparticles in superconducting structures which enhances the spin Hall effect and changes spin dynamics. Taken together, SOC in superconducting hybrid structures and the potential for electric tuning of the SOC strength, creates fascinating possibilities to advance superconducting spintronic devices for energy-efficient computing, and enable topological fault-tolerant quantum computing. By providing a description of experimental techniques and theoretical methods to study SOC, this Colloquium describes the current understanding of resulting phenomena in superconducting structures and offers a framework to select and design a growing class of materials systems where SOC plays an important role.