Abstract: Multiscale PDEs with heterogeneous, highly oscillatory coefficients pose severe challenges for standard numerical methods. Two prominent approaches to tackle such problems are numerical multiscale methods with problem-adapted coarse spaces and structured (approximate) inversion techniques that exploit a low-rank property of the associated Green’s function. They can also be used to precondition the resulting large-scale and typically very ill-conditioned linear equation systems. This work presents an abstract framework for the design and analysis of the Multiscale-Spectral Generalized FEM (MS-GFEM), a partition of unity method based on optimal local approximation spaces constructed from local eigenproblems, which is closely related to the GenEO coarse space more familiar in the DD community. We establish a general local approximation theory demonstrating, under certain assumptions, an exponential convergence w.r.t the local degrees of freedom and an explicit dependence on key parameters. Our framework applies to a broad class of multiscale PDEs with L∞-coefficients, including convection-dominated diffusion or high-frequency Helmholtz/Maxwell. Notably, we prove a nearly exponential, local convergence rate on all those problems. As a corollary, the MS-GFEM space can be used within a robust two-level DD preconditioner to achieve condition numbers arbitrarily close to one. Numerical experiments support the theoretical results and demonstrate huge efficiency gains.