The Grothendieck program for operator algebras transfers notions of Grothendieck's work from the theory of locally convex spaces and Banach spaces to the setting of C^*-algebras. The program was intensively used by Effros, Lance and Haagerup. It turns out to be particularly important for the investigation of different C^*-norms on tensor products of C^*-algebras and the analysis of approximation properties. The intention of this talk is to illustrate some more recent developments of this concept related to the local properties of C^*-algebras and non-commutative L_p-spaces.