Stability Problems in Applied Mechanics starts with the stability problems in statics. The example of buckling of columns is studied through Euler method followed by the energy method, based on Lagrange-Dirichlet theorem. Snap buckling, instability of shape, buckling due to follower load are also discussed. Insufficiency of static analysis for instability is clearly brought out and buckling problems are revisited from the point of view of dynamics. The next chapter provides the theory of Dynamical System and the foundations of bifurcation theory and explains the problems discussed in the previous chapter in the light of these unified mathematical concepts. This mathematical basis is then applied in the next chapter to investigate the stability problems encountered in dynamics of particle, rigid and flexible bodies. The last chapter explains the emergence of length scale and pattern formation as a consequence of instability in fluid, thermal and diffusion systems. Different notions of stability and the analysis of nonlinear states are briefly included in two appendices.