The main goal of this work is to initiate a program for the theory of tensor products and tensor norms in the category of operator spaces. In particular, we focus on the interplay of these theories and the theory of mapping ideals. Of course, many definitions and results in this framework are natural since they have a corresponding one in the context of Banach spaces. But many of them are not! As noticed many times, the theory is not just a straightforward translation of what is known in the classical setting and new and challenging questions naturally arise. To start with a typical difference, we highlight the role of local reflexivity, which seems to be crucial in many places as is well-known to experts. However, sometimes this hypothesis can be avoided: for example, when dealing with the class of extended \(\lambda \) -o.s. tensor norms. An unexpected issue in this category appears when relating the left accessibility of mapping ideals and associated tensor norms. The fact that their relationship is weaker than the one for its right counterpart is certainly puzzling.