The idea is quite simple. While ordinary functions assign elements of the (trivial) Hilbert space of complex numbers to a given point on the group (e.g. Euclidean space), a stochastic process associates a random variable (a non-trivial Hilbert space) to any given point. The idea of generalized functions is to think of having not a precise information at a given point, but only average values, in the complex numbers. To have a linear operator from a space of test functions into a Hilbert space is therefore the ``joint’’ generalization of those ideas. Having a suitable Banach space of test unctions which is invariant under the Fourier transform (the Segal algebra S_0(Rd)) turns out to allow for a relatively lucid and straightforward descriptions of the basic concepts relevant for the understanding of generalized stochastic processes (avoiding technical problems with vector-valued measures etc.)