  We investigate generic properties of the conformal phase transition in gauge
theories featuring Higgs-like fundamental particles. These theories provide an
excellent arena to properly investigate conformal dynamics and discover novel
features. We show that the phase transition at the boundary of the Higgs
conformal window is not smooth but a jumping one for the known perturbative
examples. In addition the general conditions under which the transition is
either jumping or smooth are provided. Jumping implies that the massive
spectrum of the theory will jump at the phase transition. It, however, still
allows for one of the states, the would be dilaton of the theory, to be lighter
than the heaviest states in the broken phase. Finally we exhibit a calculable
Higgs model in which we can, in perturbation theory, determine the Higgs
conformal window, the spectrum in the conformally broken phase and demonstrate
it to possess a jumping type conformal phase transition.
