We propose an accurate and robust Roe-type scheme applied to the compressible Euler system at all Mach numbers. To study the occurrence of unstable modes during the shock wave computation, a shock instability analysis of several Roe-type schemes is carried out. This analysis approach allows to propose a simple and effective modification to eliminate shock instability of the Roe method for hypersonic flows. A desirable feature of this modification is that it does not resort to any additional numerical dissipation on linear degenerate waves to suppress the shock instability. With an all Mach correction strategy, the modified Roe-type scheme is further extended to solve flow problems at all Mach numbers. Numerical results that are obtained for various test cases indicate that the new scheme has a good performance in terms of accuracy and robustness.