The notion that algorithmic systems should be "transparent" and "explainable" is common in the many statements of consensus principles developed by governments, companies, and advocacy organizations. But what exactly do policy and legal actors want from these technical concepts, and how do their desiderata compare with the explainability techniques developed in the machine learning literature? In hopes of better connecting the policy and technical communities, we provide case studies illustrating five ways in which algorithmic transparency and explainability have been used in policy settings: specific requirements for explanations; in nonbinding guidelines for internal governance of algorithms; in regulations applicable to highly regulated settings; in guidelines meant to increase the utility of legal liability for algorithms; and broad requirements for model and data transparency. The case studies span a spectrum from precise requirements for specific types of explanations to nonspecific requirements focused on broader notions of transparency, illustrating the diverse needs, constraints, and capacities of various policy actors and contexts. Drawing on these case studies, we discuss promising ways in which transparency and explanation could be used in policy, as well as common factors limiting policymakers' use of algorithmic explainability. We conclude with recommendations for researchers and policymakers.

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