The objective of this article is to review some developments in the use of adjoint equations in hydrodynamic stability theory. Adjoint-based sensitivity analysis finds both analytical and numerical applications much beyond those originally imagined: it can be used to identify "optimal" perturbations, or to pinpoint the most "receptive" path to breakdown, to select the most destabilizing base-flow defect in a nominally stable configuration, or to map the structural sensitivity of an oscillator. Two flow cases are focussed upon more closely in the article: the noise-amplifying instability of a boundary layer, and the global mode occurring in the wake of a cylinder. For both cases, clever interpretation and use of direct and adjoint modes provides key insight into the process of transition to turbulence.