Abstract: In this talk we discuss existing difficulties in modeling and computing energy systems and the important role that mathematical modeling and optimization can play in solving them. We argue that in some cases a change of the problem framework may be desirable and that this may be made while keeping the solution computationally achievable. We outline a number of existing and emerging fundamental research challenges and discuss some recent promising avenues in the area. A distinguishing feature of power grid applications is that optimization is ubiquitous and that it must accommodate simultaneously multiple complexity drivers. These include not only discrete variables, nonconvexity, or stochasticity but also ordinary and — with the increased usage of natural gas — partial differential equations. We discuss the productivity and performance implications of this fact for the modeling and computational environments.