The efficient production, transmission and distribution of electrical power underpins our technological civilization. Public policy and environmental concerns are leading to an increasing adoption of renewable energy sources and the deregulation of energy markets. These trends, together with an ever-growing power demand, are causing power networks to operate increasingly closer to their stability margins. Recent scientific advances in complex networks and cyber-physical systems along with the technological re-instrumentation of the grid provide promising opportunities to handle the challenges facing our future energy supply. In this thesis, we discuss the synchronization problem in power networks, which is central to their operation and functionality. We identify and exploit a close connection between the mathematical models for power networks and complex oscillator networks. Our main contributions are concise, sharp, and purely-algebraic conditions that relate synchronization in a power grid to graph-theoretical properties of the underlying electric network. Our novel conditions hold for arbitrary interconnection topologies and network parameters, and they significantly improve upon previously-available tests. We illustrate how our results help in the analysis of large-scale transmission systems and lead to novel control strategies and their implementation in microgrids. Our approach combines traditional power engineering methods, synchronization theory for coupled oscillators, and control in multi-agent dynamical systems. Beside their applications in power networks, our mathematically-appealing results are also broadly applicable in synchronization phenomena ranging from natural and life sciences to engineering disciplines.