In this paper we introduce a new theoretical paradigm, called PGR systems, which can be used to model in a discrete manner some natural phenomena occurring in-vivo/in-vitro environments. PGRsystems make use of graphs to describe the spatial structure of space of individuals, while the system dynamics caused by the movement/interaction of individuals is captured by the parallel applications of some graph rewriting rules. In this frame, an illustrative example is studied and based on it, an eloquentcomparison between the abstract rewriting machines and PGR systems is done. Several further ideas to overcome the global computational effort needed for simulations, but still maintaining the overallability for modeling are finally proposed.