This is the first talk in a series of two this semester. A non-signalling box is a matrix of conditional probabilities for a finite input- finite output experiment conducted by two experimenters, subject to the condition that `no information is exchanged'. An important result in quantum mechanics is that the non-signalling boxes that can be obtained by classical and quantum experiments are different. Tsirelson proposed two different types of quantum boxes and it is still an open problem wheter or not these two sets of matrices are the same. It is now known that Tsirelson's conjecture is equivalent to a famous conjecture of A. Connes' in von Neumann algebras. In this talk we will try to introduce some of these ideas, give fairly explicit descriptions of these various boxes in operator theory language, and indicate how these different sets of boxes can be expressed as the state spaces of various tensor products of operator systems that were introduced in our joint work with Farenick, Kavruk, Todorov and Tomforde.