AI plays an increasingly prominent role in modern society since decisions that were once made by humans are now delegated to automated systems. These systems are currently in charge of deciding bank loans, criminals' incarceration, and the hiring of new employees, and it is not hard to envision that soon they will underpin most of the society's decision infrastructure. Despite the high stakes entailed by this task, there is still a lack of formal understanding of some basic properties of such systems, including issues of fairness, accountability, and transparency. In this tutorial, we introduce a framework of causal fairness analysis, with the intent of filling in this gap, i.e., understanding, modelling, 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 in the observed data with the underlying, and often unobserved causal mechanisms that generate the disparity in the first place. We will study the problem of decomposing variations, which results in the construction of empirical measures of fairness that attribute such variations to causal mechanisms that generated them. Such attribution of disparity to specific causal mechanisms will allow us to propose a formal and practical framework for assessing legal doctrines of disparate treatment and impact, allowing also for considerations of business necessity. Finally, through the newly developed framework we will draw important connections with previous literature, both in and outside the causal inference arena. This effort will culminate in the "Fairness Map", which is the first cohesive and systematic classification device of multiple measures of fairness in terms of their causal properties, including admissibility, decomposability, and power. We hope this new set of principles, measures, and tools can help guide AI researchers and engineers when analyzing and/or developing decision-making systems that will be aligned with society's goals, expectations, and aspirations.