The chapter contains the fundamental results about Banach and quasi-Banach spaces and their complemented subspaces that are necessary for this book. Classical topics included are the Aoki-Rolewicz theorem, the completion of a quasinormed space, $p$-Banach envelopes, Pe\l czy\’nski’s decomposition method, uncomplemented subspaces of classical spaces, indecomposable spaces, type and cotype of quasi-Banach spaces, local properties, ultraproducts, the Dunford-Pettis and Grothendieck properties, properties (V) of Pe\l czy\’nski and Rosenthal, $C(K)$-spaces and their complemented subspaces and so on. More advanced topics have been also included, such as Sobczyk’s theorem and its non-separable derivatives and ultrapowers, mainly of the $L_p$-spaces.