  Relativistic O(N) field theories are studied near the quantum critical point
in two space dimensions. We compute dynamical correlations by large scale Monte
Carlo simulations and numerical analytic continuation. In the ordered side, the
scalar spectral function exhibits a universal peak at the Higgs mass. For N=3
and 4 we confirm its \omega^3 rise at low frequency. On the disordered side,
the spectral function exhibits a sharp gap. For N=2, the dynamical conductivity
rises above a threshold at the Higgs mass (density gap), in the superfluid
(Mott insulator) phase. For charged bosons, (Josephson arrays) the power law
rise above the Higgs mass, increases from two to four. Approximate
charge-vortex duality is reflected in the ratio of imaginary conductivities on
either side of the transition. We determine the critical conductivity to be
\sigma^*_c = 0.3 (\pm 0.1) 4e^2/h. In the appendices, we describe a
generalization of the worm algorithm to N>2, and also a singular value
decomposition error analysis for the numerical analytic continuation.
