Let H be an infinite-dimensional Hilbert space. A non-commutative Banach space, or operator space, is defined to be a closed linear subspace of B(H) endowed with its natural tensor product norm on K, the space of compact operators on `2. The talk will deal mainly with Banach and operator space properties of C∗- algebras, especially K itself, and non commutative L p spaces. Particular topics include operator space analogues of the separable extension property and embedding and renorming problems for non-commutative Lp spaces.