In this chapter, we present some fundamental concepts of the theory of operator spaces. While we assume familiarity with the basics, we recall certain elementary definitions for completeness. Excellent references on the topic include (Blecher and Le Merdy, Operator Algebras and Their Modules—An Operator Space Approach. London Mathematical Society Monographs. New Series, vol. 30. The Clarendon Press Oxford University Press, Oxford, 2004; Effros and Ruan, Operator Spaces. London Mathematical Society Monographs. New Series, vol. 23. The Clarendon Press Oxford University Press, New York, 2000; Paulsen, Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics, vol. 78. Cambridge University Press, Cambridge, 2002; Pisier, Introduction to Operator Space Theory. London Mathematical Society. Lecture Note Series, vol. 294. Cambridge University Press, Cambridge, 2003). The book Pisier (Tensor Products of \(C^*\) -algebras and Operator Spaces—the Connes-Kirchberg Problem. London Mathematical Society Student Texts, vol. 96. Cambridge University Press, Cambridge, 2020) also deals with tensor products of operator spaces, although concentrating on the case of \(C^*\) -algebras. Our notation follows closely that from Pisier (Astérisque (247):vi+131, 1998) and Pisier (Introduction to Operator Space Theory. London Mathematical Society. Lecture Note Series, vol. 294. Cambridge University Press, Cambridge, 2003). For technical reasons, unlike the general literature in the subject, we will not assume that our operator spaces are complete. Thus, to avoid confusion, when they are complete, we choose to call them Banach operator spaces. On the other hand, when completeness is not assumed, we refer to them as normed operator spaces or, sometimes, for simplicity, just operator spaces. The letters E, F and G will always denote normed operator spaces, that is, normed vector spaces with an additional structure at the matricial level.