In this chapter we introduce the mathematical language which is used to express the theory of General Relativity. A student coming to this subject for the first time has to become acquainted with this language, which is initially something of a challenge. Einstein himself had to learn it (from his friend Marcel Grossmann). The subject is widely known as tensor calculus; it is concerned with tensors and how to define and differentiate them in curved spaces. In more recent times tensor calculus has been recast using a more sophisticated formalism, based on coordinate-free notation and differential forms. At first physicists were disinclined to learn this higher grade language, since it involved more work, without, perhaps, any reward in terms of mathematical or physical insight. Eventually, however, sceptical minds became convinced that there were indeed pay-offs in learning this new formalism, and knowledge of it is now almost essential to read research papers in the field of General Relativity.