Local feedback stabilization of bifurcated solution branches is studied. Two cases are considered: that in which the nominal system undergoes a Hopf bifurcation as a parameter is varied, and the case of a stationary bifurcation from a simple zero eigenvalue. For each case, results on the existence of a stabilizing feedback are given. Moreover, simple synthesis techniques for the stabilizing controllers are discussed. A concept of aproximity stabilization" is introduced as an alternative to stabilization in the ordinary sense for systems that are not locally stabilizable. A result is stated on the genericity of proximity stabilizability. Motivation for further research in Beveral areas is given.