Finite state machines are fundamental computing devices at the core of many models of computation. In biology, finite state machines are commonly used as models of development in multicellular organisms. However, it remains unclear to what extent cells can remember state, how they can transition from one state to another reliably, and whether the existing parts available to the synthetic biologist are sufficient to implement specified finite state machines in living cells. Furthermore, how complex multicellular behaviors can be realized by multiple cells coordinating their states with signaling, growth, and division is not well understood. Here, we describe a method by which any finite state machine can be built using nothing more than a suitably engineered network of readily available repressing transcription factors. In particular, we show the mathematical equivalence of finite state machines with a Boolean model of gene regulatory networks. We describe how such networks can be realized with a small class of promoters and transcription factors. To demonstrate the effectiveness of our approach, we show that the behavior of the coarse grained ideal Boolean network model approximates a fine grained delay differential equation model of gene expression. Finally, we explore a framework for the design of more complex systems via an example, synthetic bacterial microcolony edge detection, that illustrates how finite state machines could be used together with cell signaling to construct novel multicellular behaviors.