We describe the main properties of the bidual of a C*-algebra, that is also its enveloping von Neumann algebra. We define the tensor norms that naturally appear on a mixed pair formed of a C*-algebra and a von Neumann one, called the nor-norm and the bin-norm. This leads naturally to the notion of local reflexivity, which, in sharp contrast with the Banach space case, is not valid for all C*-algebras. Wegive explicit examples exhibiting that phenomenon, which is specific to the min-tensor product. Indeed, we show that the analogous defect disappears for the max-tensor product.