Abstract: Among the numerical schemes for weak solutions of hyperbolic conservation laws, The nonlinear errors significantly affect the resolution of the polynomial-based high-order nonlinear weighted schemes compared with linear schemes. To mitigate this impact, researchers have systematically studied aspects such as weighted stencils, smoothness indicators, and nonlinear weighting functions. Additionally, exploratory research has been conducted on high-order nonlinear weighted schemes based on non-polynomial approaches. This paper summarizes research findings on non-polynomial WENO schemes, including trigonometric functions, logarithmic functions, radial basis functions (RBF), and hyperbolic tangent functions (THINC). It highlights their performance in terms of spectral resolution and shock-capturing capabilities. Trigonometric and RBF-based WENO schemes demonstrate superior spectral resolution compared to polynomial-based WENO schemes, while THINC-based WENO schemes exhibit significantly lower numerical dissipation when capturing discontinuities. These preliminary yet promising results reflect the new trend in the study of WENO schemes and hold significant guiding for advancing the research on high-order nonlinear weighted schemes.