This chapter is devotes to general facts and serves as an introduction to the mathematical terminology used throughout this book; it is a repertoire of results and methods to be called upon in the proofs of many theorems. It contains theorems more germane to the topic of this book and opens the way to the development. The chapter recalls all the basic mathematical tools that will be extensively employed. It aims to treat the mathematical theory of liquid crystals as a branch of modern continuum mechanics, and so it is sensible to adopt here also the mathematical structures devised to describe the sets fit to be occupied by continuous bodies and their subbodies. The chapter introduces a type of invariance which is suited for liquid crystals. It is less restrictive than isotropy and it is enjoyed by all materials which exhibit a preferred direction: it is called transverse isotropy.