Abstract: This paper describes the feasibility of a novel finite-volume based Mach-uniform semi-implicit/explicit numerical framework, MUSE for brevity, to solve unsteady compressible flows of general fluids at arbitrary Mach numbers. For high-speed flows, an explicit temporal discretization with a high-resolution upwind scheme is used so as to accurately capture shock, contact and rarefaction waves. For low-speed flows, a semi-implicit discretization with a pressure-prediction-correction method is used in order to circumvent the time step restriction arising from the Courant–Friedrichs–Lewy (CFL) condition; the high-resolution upwind scheme is used so as to preserve accuracy on the convective transport. These explicit and semi-implicit discretizations are integrated into a hybrid algorithm. In this paper, the E-CUSP scheme by G.-C. Zha is incorporated into the MUSE framework. To deal with general fluids, the GCUP method originated by K. Watanabe is introduced in a general form. Numerical experiments have been performed on a set of one-dimensional Riemann problems over a wide range of Mach numbers, from M=O(10⁻⁴) to supersonic, for a perfect gas and a liquid governed by the Tammann equation of state. The accuracy, efficiency and robustness of the proposed MUSE/E-CUSP_GCUP are discussed, and we have confirmed the validity of the method at all speeds. Smooth transition between the semi-implicit and the explicit discretizations is demonstrated in the computation of a subsonic-supersonic flow transient due to a strong local heat generation.