Abstract: Global linear stability analysis serves as a fundamental approach for investigating flow instability mechanisms, which has been extensively applied to problems such as flow past bluff bodies. This method transforms the linearized initial value problem governing the temporal evolution of infinitesimal perturbations superimposed on steady base flows into an eigenvalue problem. The solution yields direct modes with corresponding frequencies and growth rates, enabling the assessment of instability onset under given base flow conditions. While capable of characterizing exponential growth/decay behaviors of small-amplitude disturbances, this approach cannot capture stability responses under finite-amplitude forcing. Recent years have witnessed significant advancements in adjoint-based sensitivity analysis methodologies. This approach, through solving the adjoint equations, establishes sensitivity gradient relationships between eigenvalues and external input variations, thereby quantitatively evaluating the dependence of current flow stability states on external inputs. This paper systematically reviews research progress in both linear stability analysis and adjoint sensitivity analysis methods. The study first constructs theoretical frameworks for eigenvalue-equation-based linear stability analysis and adjoint-equation-based sensitivity analysis. Subsequently, it comprehensively examines four categories of external input variations affecting flow stability mechanisms: perturbed forces, base flow forces decoupled from the flow, base flow forces coupled with the flow and dimensionless parameters. Their respective sensitivity gradient characteristics are investigated to reveal, from multiple and progressively deepening perspectives, the underlying physical mechanisms and quantitative laws governing flow instability.