Traditional numerical methods can struggle to produce physically meaningful solutions or converge for nonconservative hyperbolic systems when derivatives in the source term cannot be defined in the classical sense. Such situations arise, for example, in shallow-water flow over varying topography, multiphase fluid dynamics, and flow in ducts with variable cross-sections (LeFloch, Thanh, 2023). A general framework to address this difficulty is path-conservative methods, which enable consistent approximations for the source terms (see (Castro et al., 2017)). However, selecting appropriate paths and solution admissibility criteria that accurately represent the underlying physics remain challenging, particularly for complex real-world problems, where insights from higher-order models are required. In this talk, building on the successful resolution of these challenges in the context of the Riemann problem for the shallow-water equations with a bottom discontinuity (Aleksyuk et al., 2019; 2022), I present ongoing efforts to extend these ideas to other nonconservative hyperbolic systems. Considering various physical interpretations of the nonconservative terms with undefined in the classical sense derivatives of discontinuous functions (e.g., bottom topography in shallow-water flows or cross-section area in duct flows), I highlight one that does not rely on higher-order models. For instance, in shallow-water equations, it corresponds to a piecewise-constant approximation of a continuously varying bottom, thus, allowing the Godunov method (based on the corresponding exact Riemann solver) to recover exact steady-state solutions over complex topography with coarse computational meshes (Aleksyuk, 2022). I then discuss the applicability of the admissibility criterion, proposed for the shallow-water Riemann problem in (Aleksyuk, 2019), to resolve solution nonuniqueness in other systems, potentially enabling the development of new Riemann solvers suitable for practical applications.