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The map of synaptic connectivity between neurons shapes the computations that neural circuits carry - making the identification of the design principles of neural “connectomes” crucial for understanding brain development, learning, information processing, and behavior.We present a class of probabilistic generative models for the connectomes of different brain areas in zebrafish, worm, and mouse. Our models accurately replicate a wide range of circuit properties - synapse existence and strength, neuronal in-degree and out-degree, and sub-network motif frequencies -  using surprisingly small sets of biological and physical architectural features. We then show that simulated synthetic circuits generated by our models recapitulate the neural activity and computation performed by the real ones. We extend these generative models to study the development of connectomes over time, and show they accurately replicate the “developmental trajectory” of the connectome of C. elegans, revealing a simpler set of functional cell types than commonly assumed, and identifying distinct developmental epochs. We further study structure-function relationships in simulated spiking neural networks and learn a metric that predicts the similarity of networks based on a small set of architectural features. Our findings suggest that connectomes across species follow surprisingly simple design principles and offer a general computational framework for analyzing connectomes, linking their structure to function. FOR THE LATEST UPDATES AND CONTENT ON SOFT MATTER AND BIOLOGICAL PHYSICS AT THE WEIZMANN, VISIT OUR WEBSITE: https://www.biosoftweizmann.com/