Starting from an early --- and in the light of modern theory also an abstract algebraic --- view on random variables, their algebraic and functional analytic properties emanating from simple axioms are sketched. From this abstract setting, via their connection with algebras of linear mappings and the spectral representation theory of these, one may recover Kolmogorov's classical characterisation as one particular representation. In this abstract setting a "sample" resp. "realisation" may then be identified with certain algebraic homomorphisms.
Embedded in a functional analytic framework, this allows the specification of not only all the classical spaces of random variables, but to go beyond this and and address questions of "smoothness" on the one hand, and the definition of idealised elements resp. "generalised" random variables on the other hand. This very much echoes the construction of distributions resp. generalised functions in the sense of Sobolev and Schwartz. This more abstract and general view also allows generalisations which go beyond what is possible to describe in Kolmogorov's setting, namely non-commuting random variables as they appear in quantum mechanics. Such a view is necessary to address possible novel devices like quantum computers.