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Decision-making systems based on AI and machine learning have been used throughout a wide range of real-world scenarios, including healthcare, law enforcement, education, and finance. It is no longer far-fetched to envision a future where autonomous systems will be driving entire business decisions and, more broadly, supporting large-scale decision-making infrastructure to solve society's most challenging problems. Issues of unfairness and discrimination are pervasive when decisions are being made by humans, and remain (or are potentially amplified) when decisions are made using machines with little transparency, accountability, and fairness. In this paper, we introduce a framework for \textit{causal fairness analysis} with the intent of filling in this gap, i.e., understanding, modeling, and possibly solving issues of fairness in decision-making settings. The main insight of our approach will be to link the quantification of the disparities present on the observed data with the underlying, and often unobserved, collection of causal mechanisms that generate the disparity in the first place, challenge we call the Fundamental Problem of Causal Fairness Analysis (FPCFA). In order to solve the FPCFA, we study the problem of decomposing variations and empirical measures of fairness that attribute such variations to structural mechanisms and different units of the population. Our effort culminates in the Fairness Map, which is the first systematic attempt to organize and explain the relationship between different criteria found in the literature. Finally, we study which causal assumptions are minimally needed for performing causal fairness analysis and propose a Fairness Cookbook, which allows data scientists to assess the existence of disparate impact and disparate treatment.