We focus on the tensor products of operator systems and examine several examples including minimal, maximal, maximal commuting, and some asymmetric tensor products. We refine the notion of nuclearity in this setting by studying operator systems that preserve various pairs of tensor products. As an application of this theory we discuss some new equivalences of the Kirchberg Conjecture. This presentation is based on joint work with V.I. Paulsen, I.G. Todorov and M. Tomforde.