Trefftz methods are a class of high-order numerical schemes for solving problems governed by partial differential equations (PDEs), and have been studied extensively for wave propagation problems. They construct the numerical solution in spaces of functions locally tailored to the PDE, with test and trial functions that are exact solutions of the governing equation on each mesh element. Notably, Trefftz methods can achieve high accuracy at reduced computational cost compared to traditional approaches such as finite element or standard discontinuous Galerkin methods. For wave propagation in inhomogeneous media, material heterogeneities are modeled by variable coefficients in the governing PDE. However, the application of Trefftz methods is typically limited to linear PDEs with homogeneous, piecewise-constant coefficients, since exact local solutions are generally unavailable in more complex settings. To overcome this limitation, quasi-Trefftz methods employ element-wise approximate solutions of the PDE, allowing the study of a broader class of problems while preserving many favorable properties of Trefftz schemes. Recent studies have established convergence and stability results for quasi-Trefftz methods applied to selected scalar problems, such as the diffusion–advection–reaction equation. This talk focuses on the analysis of a class of discrete spaces within the quasi-Trefftz framework, namely polynomial quasi-Trefftz spaces. We present recent results and new insights aimed at extending quasi-Trefftz methods to complex vector-valued PDEs, in particular the time-harmonic Maxwell equations. After introducing the quasi-Trefftz approach and the model problem, we propose a novel family of polynomial quasi-Trefftz spaces for time-harmonic Maxwell equations with smooth variable coefficients and analyze their approximation properties. An efficient algorithm for constructing bases of these spaces is presented, followed by numerical experiments illustrating the theoretical findings.