In order for computation to emerge spontaneously and become an important factor in the dynamics of a system, the material substrate must support the primitive functions required for computation: the transmission, storage, and modification of information. Under what conditions might we expect physical systems to support such computational primitives This paper presents research on Cellular Automata which suggests that the optimal conditions for the support of information transmission, storage, and modification, are achieved in the vicinity of a phase transition. We observe surprising similarities between the behaviors of computations and systems near phase-transitions, finding analogs of computational complexity classes and the Halting problem within the phenomenology of phase-transitions. We conclude that there is a fundamental connection between computation and phase-transitions, and discuss some of the implications for our understanding of nature if such a connection is borne out. 31 refs., 16 figs.