Fluid flows subject to time-dependent external forces or boundary conditions are ubiquitous in biology and technical applications. Whether one considers birds flying by flapping their wings or the pumping of blood through our arteries, the flow is non-autonomous. We investigate the influence of external time-dependence on the non-linear evolution of fluid flows, as well as on the linear response to small disturbances that determines their stability. For the analysis of the time-periodic pulsatile flow through toroidal pipes, an iterative fixed-point solver in frequency space is developed and validated to obtain the baseflows. The method is used to explore the effect of pulsations on the flow through tori with relevant curvatures. Using the Floquet framework, the linear stability of the flow close to criticality is investigated, revealing strong sensitivity to pulsations that are mostly stabilising. Considering the local stability of pulsating plane Poiseuille flow, the eigenpairs of the linear operator are tracked over time producing subharmonic eigenvalue orbits. Their appearance is traced to spectral degeneracies of the operator, leading to the transition of the harmonic disturbance between eigenvalue trajectories involving non-modal growth bursts. The same flow case is then used to assess the potential of the optimally time-dependent (OTD) framework for transient linear stability analysis of flows with arbitrary time-dependence using a localised linear/non-linear implementation aimed at open flows. This transient linear stability framework is then extended to three-dimensional flows and used to track the linear stability of a laminar separation bubble on a pitching wing section. On a natural laminar flow airfoil exhibiting an abrupt change of the transition location during the pitching motion, the appearance of a global mode corresponding to an absolute local instability is identified at the rear of the bubble which is linked to its sudden breakdown to turbulence.