The book focuses on the basic theory of nonlocal and fractional operators and it is divided into two main parts covering respectively the basic theory of the "singular-integral" fractional Laplacian operator, which also includes a number of explicit examples of its evaluations, and the regularity theory in L^p spaces, by going through Bessel and Besov spaces. A number of appendices then go over additional material in order to make the exposition as self-contained as possible. The book is suitable for a PhD course but can be also read by advanced undergraduates. It contains didactic elements, but it is also useful for researchers, since it collects essential material in a concise and self-constrained way.