The Holomorphic Embedding Load-Flow Method (HELM) was recently introduced as a novel technique to constructively solve the power flow equations in power networks, based on advanced concepts from complex analysis, algebraic curves, and modern techniques in approximation theory. HELM's results are always guaranteed and unequivocal: if the power flow problem is feasible, it constructs the most desirable solution; and conversely, if the power flow problem is infeasible, it signals such condition reliably. This monograph provides the first in-depth and comprehensive overview of HELM. Starting by revisiting the theoretical foundations, it proceeds to develop the complete mathematical basis required to develop practical solutions to the problem. These formulations cover both ac and dc networks. The significance of HELM extends beyond its utilitarian role as a reliable power flow solver, since the theory backing this method is proving to be a fertile ground for the development of new analysis tools for power systems. This monograph is a must-read for all students and researchers working on the cutting edge of electric power systems.