The chapter presents a collection of essential facts and definitions frequently used throughout the text. We recall the definitions of the Skorokhod \(J_1\) -topology on the space of càdlàg functions, provide a brief overview of stable distributions and their domains of attraction, and discuss generalized inverse functions and their properties. Additionally, we review fundamental results on weak convergence of probability measures, including the continuous mapping theorem, the Skorokhod representation theorem, Donsker’s invariance principle, and convergence of Markov processes. Towards the end of the chapter, we delve into Itô’s excursion theory, a more advanced topic which is instrumental in our construction of a skew stable Lévy process. Typically, Itô’s excursion theory is not covered in standard textbooks.