The electrical power grid (the electricity transmission and distribution system) is one of the greatest and most complex engineering achievements of the 20th century. However, it is also at the center of massive changes in the way we create and consume energy that are brought about by many drivers, including an increasing usage of renewable energy and natural gas. Moreover, it exhibits persistent conceptual difficulties that, while generally successfully contained by engineering practice, have never been fully resolved. In this talk we discuss some of these challenges and the important role that mathematical modeling and optimization can play to solve them. We will argue that in some cases a change of the problem framework may be desirable and that this may be done while keeping the solution computationally achievable. We will 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, non convexity or stochasticity, but also ordinary and, with the increased usage of natural gas, partial, differential equations. We will discuss the productivity and performance implications of this fact for the modeling and computational environments. I have been a Senior Computational Mathematician in the Mathematics and Computer Science Division at Argonne National Laboratory since 2013, and a Computational Mathematician between 2002-2013. I have been a Professor with tenure in the Department of Statistics at the University of Chicago since 2012, jointly appointed with Argonne. Previously, I had been a part-time Professor of Statistics at the University of Chicago since 2009. Between 1999 and 2002 I was an Assistant Professor of Mathematics at the University of Pittsburgh. From 1997 to 1999 I was the Wilkinson Fellow in scientific computing at Argonne. I am currently an associate editor for Mathematical Programming series A and B, SIAM Journal on Optimization, SIAM Journal in Scientific Computing, SIAM-ASA Journal on Uncertainty Quantification and a software editor for Optimization Methods and Software. I specialize in numerical optimization, numerical analysis, and uncertainty quantification. I have advised two Ph.D students at Pitt, two Masters students at the University of Chicago, 22 summer interns at Argonne, and 18 postdoctoral scholars at Argonne. I have sponsored, supervised and mentored 6 full-time scientific employees at Argonne (5 junior scientists and one predoc), all of which hold currently scientific positions or are enrolled in Ph.D programs. I have been a lead investigator in several competitively awarded grants whose total funding amount exceeds 20 million dollars. My work has been primarily in the are of numerical optimization, numerical analysis and uncertainty quantification and their applications in areas related to energy.