In this chapter, we walk the reader through the different components of the Collections Directory. We begin with a discussion of the mathematical fundamentals, for which we use the book by Karniadakis and Sherwin [48] as our principle reference. We then provide the reader with an overview of the primary data structures introduced within the Collections Directory (often done through C++ objects), and then present the major algorithms – expressed as either object methods or functions – employed over these data structures.