This work deals with various aspects of boundary-layer stability. Modal and non-modal approaches are first used in the study of the global stability of a jet in crossflow. This flow case presents a global instability in some regimes which results from a Hopf bifurcation from a steady wake to a limit cycle consisting of a shedding of hairpin vortices. The effects of non-normality are studied in relation with transient growth and numerical accuracy. It is shown that the equations must be solved to a very high accuracy in order to properly capture the spectrum and that the computational domain must be very long due to the elongated core of the instability. Non-modal techniques do not suffer from such issues. The so-called acoustic receptivity of a flat plate with a leading-edge is analysed using a global modes approach. This leads to a spatio-temporal analysis in which the modes must be corrected for the imaginary part of the eigenvalues. This correction involves the Parabolised Stability Equations (PSE). This work confirms results previously obtained through different methods. The stability of two- and three-dimensional boundary-layer flows in the presence of surface irregularities such as steps, gaps or humps is also studied using Direct Numerical Simulation (DNS). It is found that all the surface irregularities have a destabilising effect on stability of two-dimensional boundary layers, with the rectangular hump case being the most dangerous one.  In the case of three-dimensional boundary layers the effects are more complex. Our results accurately reproduce the steady flows, caused by small  forward-facing steps, from an experimental setup, and the interaction of saturated crossflow vortices with unsteady noise is discussed. This work also describes a new method related to modal decomposition of compressible flows with shocks. Traditional linear techniques such as the Proper Orthogonal Decomposition (POD) struggle to capture strong nonlinear phenomena such as shock motion.  The proposed shock-fitting approach tackles this issue by interpolating data onto a grid following the discontinuities. This requires detecting and parametrising the shocks, then mapping the original flow fields onto a reference mesh. A method to generate this mapping in two-dimensional domains is presented. Then the method is applied to two two-dimensional cases in ascending complexity. In addition to faster decay of the singular values, the modes obtained are cleaner and devoid of oscillations around the shocks.