Galactic structure is generally studied from the point of view of Newtonian dynamics. The goal of this paper is to show that modern statistical physics also has an important contribution to make in understanding the structure of galaxies. In particular, we show how the process of star formation can be modelled as a percolation process, and how the phase transition associated with this percolation process plays a critical role in the stabilization and control of star formation. For galaxies well described by our model, the dominant morphological feature, the presence of spiral arms, is a consequence of the proximity to the second-order phase transition associated with the percolation threshold. We present an introduction to the astrophysical problem we are investigating and an analysis of the directed percolation problem, called stochastic self-propagating star formation, that controls star formation. We also describe a simulation of the galactic evolution problem in terms of a cellular automaton that provides a representation of the percolation process of interest in an astrophysical environment. Lastly, we provide a number of examples of how the model gives a good account of much of the observed nature of spiral galaxies. © Taylor & Francis Group, LLC.