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We  first describe the mathematical foundation of DSGRN (Dynamic Signatures Generated by Regulatory Networks), an approach that provides a combinatorial description of global dynamics of a network over its  entire parameter space. Finite description allows comparison of parameterized dynamics between hundreds of networks  to  discard networks that are not compatible with experimental data. We  describe a  close connection of DSGRN  to Boolean network models that allows us to view DSGRN as a platform for bifurcation theory of Boolean maps. Finally, we describe current efforts to rigorously connect the combinatorial structures to structure of attractors of continuous ODE models of network dynamics. We discuss  several applications of this methodology  to systems biology.