Abstract: We determine completely the exact Riemann solutions for the system of Euler equations in a duct with discontinuous varying cross-section. The crucial point in solving the Riemann problem for hyperbolic system is the construction of the wave curves. To address the difficulty in the construction due to the nonstrict hyperbolicity of the underlying system, we introduce the L–M and R–M curves in the velocity–pressure phase plane. The behaviors of the L–M and R–M curves for six basic cases are fully analyzed. Furthermore, we observe that in certain cases the L–M and R–M curves contain the bifurcation which leads to the non–uniqueness of the Riemann solutions. Nevertheless, all possible Riemann solutions including classical as well as resonant solutions are solved in a uniform framework for any given initial data.