PAC-Bayesian analysis is a basic and very general tool for data-dependent analysis in machine learning. By now, it has been applied in such diverse areas as supervised learning, unsupervised learning, and reinforcement learning, leading to state-of-the-art algorithms and accompanying generalization bounds. PAC-Bayesian analysis, in a sense, takes the best out of Bayesian methods and PAC learning and puts it together: (1) it provides an easy way to exploit prior knowledge (like Bayesian methods); (2) it provides strict and explicit generalization guarantees (like VC theory); and (3) it is data-dependent and provides an easy and strict way of exploiting benign conditions (like Rademacher complexities). In addition, PAC-Bayesian bounds directly lead to efficient learning algorithms. Thus, it is a key and basic subject for machine learning. While the first papers on PAC-Bayesian analysis were not easy to read, subsequent simplifications made it possible to explain it literally in three slides. We will start with a general introduction to PAC-Bayesian analysis, which should be accessible to an average student, who is familiar with machine learning at the basic level. Then, we will survey multiple forms of PAC-Bayesian bounds and their numerous applications in different fields, including supervised and unsupervised learning, finite and continuous domains, and the very recent extension to martingales and reinforcement learning. Some of these applications will be explained in more details, while others will be surveyed at a high level. We will also describe the relations and distinctions between PAC-Bayesian analysis, Bayesian learning, VC theory, and Rademacher complexities. We will discuss the role, value, and shortcomings of frequentist bounds that are inspired by Bayesian analysis.