The purpose of this textbook is to present: an array of topics that are found in the syllabus of the typical second lecture course in Calculus, as offered in most Italian universities. Conceptually, it follows our previous book Mathematical Analysis I, published by Springer, which will be referred to throughout as Vol. I. While the subject matter known as ‘Calculus 1’ concerns real functions of real variables, and as such is more or less standard, the choices for a course on ‘Calculus 2’ can vary a lot, and even the way the topics can be taught is not so rigid. Due to this larger flexibility we tried to cover a wide range of subjects, reflected in the fact that the amount of content gathered here is not even slightly comparable to the number of credits conferred to a second Calculus course by the current programme specifications.