The solution set of a nonlinear problem may be very complicated and may change dramatically under a small perturbation, when the qualitative behavior of the natural phenomenon it models lacks stability. In particular, some branching and bifurcating solutions may appear. The object of bifurcation theory is precisely to study when there is a “branching” of new solutions. Bifurcation is a very important subject in modern nonlinear analysis that dates from the nineteenth century and to which many books and a multitude of papers have been devoted. It has found applications in many areas including elasticity theory, fluid dynamics, geophysics, astrophysics, meteorology, statistical mechanics, chemical kinetics, and so on. The mathematical side is also very rich. Bifurcation problems have been treated using many approaches: methods of function theory, algebra, algebraic geometry, critical point theory (both Morse and Ljusternik-Schnirelman approaches), algebraic topology, and differential topology. Nevertheless, the simplest cases may be solved using only the implicit function theorem.