In this introduction, we provide an overview of the main results proven in this book, along with some fundamental concepts. Let X = G/H be a reductive homogeneous space with noncompact H, endowed with a G-invariant pseudo-Riemannian structure. Our focus in this book is on spectral analysis on standard pseudo-Riemannian locally homogeneous spaces XΓ = Γ\G/H, called standard, in the setting that Γ is a torsion-free discrete subgroup of a reductive subgroup L of G acting properly and spherically on X. We provide a complete list of such triples of Lie groups (G, H, L) infinitesimally. Examples of such G/H include anti-de Sitter spaces SO(2n, 2)/SO(2n, 1), indefinite Kähler manifolds SO(2n, 2)/U(n, 1), and the non-symmetric reductive homogeneous space SO(4, 3)/G2(2). Compared to the traditional Riemannian context where H is compact, we encounter new challenges: L2 (XΓ) is no longer a subspace of L2 (Γ\G), on which the group G acts as a unitary representation. Moreover, it is unclear whether the pseudo-Riemannian Laplacian is essentially self-adjoint, in the absence of a general theory. We formulate our main results regarding the spectral decomposition of compactly supported smooth functions into joint eigenfunctions, the abundance of real analytic joint eigenfunctions, and the existence of an infinite L2 spectrum under certain additional conditions.