A survey is given for some recent developments on bifurcation from local steady-states to global dynamics governed by nonlinear ODE's with Z2 or O(2) symmetries. In particular, we are mainly concerned with a double singular point, which is a Z2 symmetric steady-state possessing a Jacobian with two zero eigenvalues. There exist, under suitable con-ditions, Hopf points and heteroclinic cycles bifurcating from the double singular point. We also considered a triple zero point with an O(2) sym-metry, from which bifurcate standing waves, rotating waves and modu-lated rotating waves.