This paper is concerned with the problem of finding a universal dynamical system for all dynamical systems on separable metric spaces. Special care is given to exhibit a universal dynamical system which was used to motivate the definition of a dynamical system. We establish that this class of dynamical systems is topologically as narrow as a system describable by a first-order partial differential equation. We find that a classical solution space of this partial differential equation will serve as the phase space of a universal system for dynamical systems on locally compact separable metric spaces. In fact, the functions in this solution space areC ∞ and vanish at infinity. For the remaining dynamical systems on separable metric spaces we find a universal system similar to the shift system exhibited by Bebutov. The marked difference is that there is no restriction on the set of rest points. Further comments concerning the history of this problem follow some basic definitions given in the introduction.