  We present models to predict high frequency counts of extragalactic radio
sources using physically grounded recipes to describe the complex spectral
behaviour of blazars, that dominate the mm-wave counts at bright flux
densities. We show that simple power-law spectra are ruled out by
high-frequency (nu>100 GHz) data. These data also strongly constrain models
featuring the spectral breaks predicted by classical physical models for the
synchrotron emission produced in jets of blazars (Blandford & Konigl 1979;
Konigl 1981). A model dealing with blazars as a single population is, at best,
only marginally consistent with data coming from current surveys at high radio
frequencies. Our most successful model assumes different distributions of break
frequencies, nu_M, for BL Lacs and Flat-Spectrum Radio Quasars (FSRQs). The
former objects have substantially higher values of nu_M, implying that the
synchrotron emission comes from more compact regions; therefore, a substantial
increase of the BL Lac fraction at high radio frequencies and at bright flux
densities is predicted. Remarkably, our best model is able to give a very good
fit to all the observed data on number counts and on distributions of spectral
indices of extragalactic radio sources at frequencies above 5 and up to 220
GHz. Predictions for the forthcoming sub-mm blazar counts from Planck, at the
highest HFI frequencies, and from Herschel surveys are also presented.
