The objective of this paper is to present a general study of the stability and bifurcation of elastic structures in the presence of unilateral contact with or without friction. The stability in the dynamic sense of an equilibrium position is first considered for a contact without friction. It is shown that a local strict minimum of the potential energy is a stable equilibrium. The positivity of the second variation of the lagrangean with respect to some compatible virtual displacement rates is a criterion for dynamic stability. For a contact with Coulomb friction, the study of the rate problem leads to a non-bifurcation criterion and to a stability criterion as in plasticity.