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This book, the first volume of a two-part monograph, is centered on key background fundamentals and results, and on the description of the State- and Property-Specific Approach (SPSA) to the construction and implementation of efficacious methods for the quantitative solution of various types of time-independent or time-dependent problems in Quantum Chemistry and in Atomic, Molecular, Optical, and Chemical Physics. By going beyond the standard many-electron problem for ground states, the discussion mainly addresses the key aspects of the SPSA methodology for the calculation of wavefunctions of excited discrete states and of resonance (autoionizing) states that are tailored to each property or phenomenon of interest. In particular, the SPSA uses state-specific function spaces and practicable computational methods that allow insightful and economic descriptions of excited-state electronic structures and the systematic calculation of the interplay between electronic structures (including electron correlations) on the one hand and spectra and dynamics on the other. The Hamiltonians (nonrelativistic or relativistic) are either field-free or include weak or strong electromagnetic fields that may be static, or periodic, or pulsed. The arguments and commentary in both volumes of the monograph are supported by a plethora of numerical examples and by comparisons with experiment.