This is the first volume in a comprehensive two-volume set, 'Advances in Quantum Science', whose second volume is 'Advances in Quantum Science II: Computational and Experimental Approaches'. This volume presents novel quantum theories and computational techniques designed to advance cutting-edge research. It commences with an introduction to updated theories of the quantum electrodynamics and density matrix. Electrons and positrons are examined from diverse theoretical perspectives. Quantum entanglement is explained from the basics to quantum scattering. In Part II, newly developed computational methods are explored. Ultrafast multistage charge transfer in molecular systems is examined with considerations of complex relaxation dynamics. The text delves into computational strategies for managing exciton storage in quantum batteries.  The integration of relativistic effects into the coupled cluster method is introduced. This book aims to assist researchers engaged with electron studies in discovering insights to initiate pioneering research.