This paper aims at reviewing linear and nonlinear approaches to study the stability offluid flows. We provide a concise but self-contained exposition of the main concepts andspecific numerical methods designed for global stability studies, including the classicallinear stability analysis, the adjoint-based sensitivity, and the most recent nonlineardevelopments. Regarding numerical implementation, a number of ideas making resolu-tion particularly efficient are discussed, including mesh adaptation, simple shift-invertstrategy instead of the classical Arnoldi algorithm, and a simplification of the recent non-linear self-consistent (SC) approach proposed by Manticˇ-Lugo et al. (2014, “Self-Consistent Mean Flow Description of the Nonlinear Saturation of the Vortex Shedding inthe Cylinder Wake,” Phys. Rev. Lett., 113(8), p. 084501). An open-source software imple-menting all the concepts discussed in this paper is provided. The software is demon-strated for the reference case of the two-dimensional (2D) flow around a circularcylinder, in both incompressible and compressible cases, but is easily customizable to avariety of other flow configurations or flow equations.