  We present the universal theory of arbitrary, localized impurities in a
confining paramagnetic state of two-dimensional antiferromagnets with global
SU(2) spin symmetry. The energy gap of the host antiferromagnet to spin-1
excitations, \Delta, is assumed to be significantly smaller than a typical
nearest neighbor exchange. In the absence of impurities, it was argued in
earlier work (Chubukov et al. cond-mat/9304046) that the low-temperature
quantum dynamics is universally and completely determined by the values of
\Delta and a spin-wave velocity c. Here we establish the remarkable fact that
no additional parameters are necessary for an antiferromagnet with a dilute
concentration of impurities, n_{imp} - each impurity is completely
characterized by a integer/half-odd-integer valued spin, S, which measures the
net uncompensated Berry phase due to spin precession in its vicinity. We
compute the impurity-induced damping of the spin-1 collective mode of the
antiferromagnet: the damping occurs on an energy scale \Gamma= n_{imp} (\hbar
c)^2/\Delta, and we predict a universal, asymmetric lineshape for the
collective mode peak. We argue that, under suitable conditions, our results
apply unchanged (or in some cases, with minor modifications) to d-wave
superconductors, and compare them to recent neutron scattering experiments on
YBCO by Fong et al. (cond-mat/9812047). We also describe the universal
evolution of numerous measurable correlations as the host antiferromagnet
undergoes a quantum phase transition to a Neel ordered state.
