Objectives: Discussions of fairness in criminal justice risk assessments typically lack conceptual precision. Rhetoric too often substitutes for careful analysis. In this paper, we seek to clarify the tradeoffs between different kinds of fairness and between fairness and accuracy. Methods: We draw on the existing literatures in criminology, computer science and statistics to provide an integrated examination of fairness and accuracy in criminal justice risk assessments. We also provide an empirical illustration using data from arraignments. Results: We show that there are at least six kinds of fairness, some of which are incompatible with one another and with accuracy. Conclusions: Except in trivial cases, it is impossible to maximize accuracy and fairness at the same time, and impossible simultaneously to satisfy all kinds of fairness. In practice, a major complication is different base rates across different legally protected groups. There is a need to consider challenging tradeoffs.