Recent fire-related damages and fatalities caused by high-voltage transmission lines coupled with dry weather are costing billions of dollars annually, with the only practical solution being de-energizing the lines and disrupting the power supply to millions of customers. The recent advances in the distribution grid, including the integration of distributed generation (DGs), distributed energy resources (DERs), and microgrids, provide potential means to improve the grid's operational resilience. An advanced decision-support system is needed to plan and manage grid operations by proactively managing the grid's variable, uncertain, and distributed resources. Consequently, resilient operational solutions for power distribution grids have drawn significant attention. These applications range from leveraging recent advances in smart grid technology, such as remote control capabilities and DER integration, to enabling advanced grid services such as frequency and voltage support for the bulk grid and resilient operations through intentional islanding to support critical services during disruptions. This monograph provides a much-needed primer on optimization methods used in active power distribution systems for advanced operations, with the goal of benefiting researchers working in this field. The graduate students and young researchers working in the area of DERs and distribution systems operations need a background on not only topics related to power distribution engineering but also a wide variety of interdisciplinary subjects to address the upcoming challenges. The monograph will benefit a diverse pool of researchers and industry practitioners by building the necessary background on modeling the distribution systems (with DERs) and system optimization methods for provisioning grid services.