After a brief discussion of quantization in physics, we will discuss the analogous program in mathematics---mathematical quantization. We will then explain how mathematical quantization applied to the theory of Banach spaces leads to the theory of operator spaces. In the second half of the talk, we will discuss three areas of operator space theory to which we have made contributions:

(1) One-sided multipliers and M-ideals of operator spaces. (2) Algorithms for deciding complete positivity. (3) Arveson's non-commutative Choquet boundary.