<p>Robust and accurate numerical methods for numerical simulations of compressible fluid flows featuring discontinuities around complex geometries are challenging and of increasing practical interest. In this paper, a third-order weighted compact nonlinear scheme (WCNS) for solving inviscid compressible flows on Cartesian grids and a third-order constrained weighted least squares immersed boundary (CWLS-IB) method for dealing with irregular boundaries are developed. The numerical fluxes at the cell edges are calculated by a nonlinear combination of an optimal high-order polynomial on a global stencil and two low-order polynomials on two sub-stencils. A new global smoothness indicator of the global stencil is introduced to improve the accuracy of the scheme at both the first- and second-order critical points. The third-order CWLS method is employed to extrapolate the values at ghost cells outside the physical domain and a robust WENO-type extrapolation is adopted to eliminate oscillations when shocks occur near boundaries by introducing a nonlinear weight that can switch automatically between the high-order and the low-order extrapolation. Several numerical experiments are provided to verify the performance of the present WCNS scheme and the CWLS-IB method in terms of the accuracy, robustness and spatial resolution.</p>