This thesis concerns with hydrodynamic stability of fluid flows. Direct numerical simulation (DNS) is used to investigate the non-linear dynamics of the flow and to obtain the basic states. We develop a new procedure (named BoostConv) able to stabilize the dynamical system without nega- tively impacting on the computational time of the original numerical proce- dure. The stability and transition of several flow configurations, such as the flow over an open cavity, the flow past a sphere or a hemispherical roughness element are investigated. In particular, a modal stability analysis is used to study the occurrence of possible bifurcations. Both direct and adjoint eigen- modes are considered and the region of the flow responsible for causing the global instability is identified by the structural sensitivity map. Moreover, we generalize the latter concept by including second-order terms. We apply the proposed approach to a confined wake and show how it is possible to take into account the spanwise wavy base-flow modifications to control the instability. Inspired by the sensitivity field obtained to localize the ’wave- maker’ in complex flows, we introduce the Error Sensitivity to Refinement (ESR) suitable for an optimal grid refinement that minimizes the global so- lution error. The new criterion is derived from the properties of the adjoint operator and provides a map of the sensitivity of the global error (or its estimate) to a local mesh adaptation. Finally, we investigate the stability of unsteady boundary layers using the complex-ray theory. This theory allows us to describe the propagation of small disturbances by a high-frequency (optical) approximation similar to the one adopted for wave propagation in nonuniform media. [edited by Author]