In this book, we adopt a new approach to spectral analysis on standard pseudo-Riemannian locally homogeneous spaces XΓ = Γ\G/H, beyond the conventional Riemannian setting where H is compact. This chapter presents an overview of our methodology. Additionally, we give some refinements to the results outlined in Chapter 1 . The remainder of the book primarily focuses on proving these results. Suppose a reductive subgroup L of G acts properly on X = G/H, and Γ is a torsion-free discrete subgroup of L. Our strategy for spectral analysis on the standard quotient XΓ involves employing Γ-periodic joint eigenfunctions for differential operators derived from two different sources: In general, these two sources (1) and (2) are not closely linked, primarily because the branching laws for infinite-dimensional representations for the restriction G ↓ L are not well-behaved, due to the presence of continuous spectrum or infinite multiplicities. However, if L acts properly and spherically, the branching laws become discretely decomposable, and multiplicities are uniformly bounded. In this case the two sources (1) and (2) are closely linked through transfer maps, as we describe in this chapter.