Suppose (Ω i , F i ), 1 ≤ i ≤ n are sample spaces describing the elementary outcomes and events concerning n different statistical systems in classical probability. To integrate them into a unified picture under the umbrella of a single sample space one takes their cartesian product (Ω, F) where Ω = Ω1 x … x Ω n , F = F1 x … x F n , the smallest σ-algebra containing all rectangles of the form F1 x F1 x … F n , F j ∈ F j for each j. Now we wish to search for an analogue of this description in quantum probability when we have n systems where the events concerning the j-th system are described by the set P(ℋ j ) of all projections in a Hubert space ℋ j j = 1, 2,…, n. Such an attempt leads us to consider tensor products of Hilbert spaces. We shall present a somewhat statistically oriented approach to the definition of tensor products which is at the same time coordinate free in character. To this end we introduce the notion of a positive definite kernel.