This paper is a general course on the analysis of stability and bifurcation for dissipative systems. It provides an unified description for the study of the quasi-static response and of the analysis of stability and bifurcation in a variety of interesting applications. The theoretical presentation is complemented by a number of simple analytical examples. The first main issue to be addressed pertains to the buckling of elastoic-plastic structures. Stability and non-bifurcation criteria are given for the generalized standard models of plasticity in the light of Hill's discussions. The second main issue to be addressed pertains to the stability analysis for a solid with internal damage or multiple cracks. In brittle fracture or brittle damage, the evolving equations for the crack lengths or for the damage papaneters are time-independent as in plasticity and leads to similar mathematical considerations on stability and bifurcation analysis.