Abstract: Power grids are almost universally agreed to be complex systems, which means that it is not possible to fully understand the grid by just looking at its parts. Power grids, which we define here to include all of the physical infrastructure and human individuals and organizations that jointly work to produce, distribute and consume electricity, have many properties that are common to other complex systems. Like the international financial system, power grids are operated by many millions of physical (hardware/software) and human agents. Like the Internet, power systems are frequently subjected to both random failure and malicious attack. Like the weather systems interacting to form hurricanes, there are strong, non-linear connections among the components, and between the components and society at large. And power systems occasionally exhibit spectacularly large, and costly, failures. This essay attempts to help us to understand these failures by highlighting key mathematical properties of cascading failures in complex systems in general, and in power grids in particular. We focus particularly on the mathematical challenges of measuring cascading failure risk in large power grids, and discuss some techniques that may provide better information to power grid operators regarding cascading failure risk.