Recall that given \(C \ge 1\) , a normed operator space is said to have the C-completely bounded approximation property (C-CBAP for short) if there exists a net of finite-rank mappings \(\phi _i : E \to E\) such that \( \left \|\phi _i\right \| _{ \operatorname {\mathrm {cb}}} \le C\) for all i and for every \(x \in E\) , \( \left \|\phi _i(x) - x\right \| \to 0\) . One of the goals in this chapter is to study the CBAP via various conditions involving tensor products, in the spirit of the characterizations for the operator space approximation property appearing in Effros and Ruan (Operator Spaces. London Mathematical Society. Monographs. New Series, vol. 23. The Clarendon Press Oxford University Press, New York, 2000, Sec. 11.2). We also deal with a weaker version, called W*CBAP, which carries over many of the interesting equivalences that appear in the classical theory of tensor products (Defant and Floret, Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176. North-Holland Publishing Co., Amsterdam, 1993, Sec. 16).