The development of electronic structure methods is essential for understanding the properties of molecules and materials, thereby advancing materials and drug discovery. Over the past century, numerous electronic structure methods have been developed, though the most accurate ones often suffer from poor scalability with system size, limiting their application to moderate-sized molecules and extended materials. Quantum embedding offers a promising solution by fragmenting the system into smaller, chemically relevant pieces and applying the accurate, computationally expensive methods only on the chemically important fragments. In this thesis, we discuss the development and application of quantum embedding methods that significantly reduce computational costs, in terms of computational time and memory, using both classical and quantum computing resources. Using illustrative examples, we show that these approaches make certain calculations possible that were previously impractical with conventional electronic structure methods. Chapter 1 of this thesis presents a review of the fundamentals of electronic structure theory relevant to this work and provides a general outline for the rest of the thesis. In chapters 2 and 4, we will discuss methods developed using density matrix embedding techniques and strongly correlated electronic structure theories to study excited states of defects in extended solids, which are essential for the discovery of solid-state qubits and in heterogeneous catalysis applications. In chapter 3, we extend our quantum embedding framework with weakly correlated electronic structure theories to study surface adsorption through which we develop memory-efficient techniques that lead to savings in computational memory of handling larger fragment sizes. In chapter 5, we develop a quantum computing algorithm based on the localized active space self-consistent field (LASSCF) method that significantly reduces resource requirements for quantum computers as needed for application on noisy intermediate-scale quantum devices. Finally, in chapter 6, we present an application of multireference methods in studying metal-metal bonding in uranium dimers. These methods enable more extensive calculations on moderate to large molecules and materials, providing a framework for the future development of accurate electronic structure methods at a manageable cost.