The double coset space Γ\G/H does not admit a nontrivial group action in general. To relate spectral analysis for intrinsic differential operators on Γ\G/H with representation theory, we regard functions on the locally homogeneous space Γ\G/H as Γ-periodic functions on the homogeneous space G/H or as H-invariant functions on Γ\G. Our approach to spectral analysis on standard pseudo-Riemannian locally homogeneous spaces Γ\G/H involves comparing two homogeneous spaces: one being the homogeneous space G/H, and the other a nonsymmetric space possessing a favorable property, namely, a real form of a spherical space. In this chapter, we provide an overview of key results concerning real spherical spaces and real forms of spherical spaces beyond symmetric spaces, laying the groundwork for their application in this book. Notably, in the former spaces, the multiplicities of all irreducible representations are finite, while in the latter, these multiplicities are uniformly bounded, and the ring of differential operators is commutative. These properties will form the foundational structure for the later chapters’ proof of spectral analysis on the locally homogeneous spaces Γ\G/H.