  We present the general theory of clean, two-dimensional, quantum Heisenberg
antiferromagnets which are close to the zero-temperature quantum transition
between ground states with and without long-range N\'{e}el order. For
N\'{e}el-ordered states, `nearly-critical' means that the ground state
spin-stiffness, $\rho_s$, satisfies $\rho_s \ll J$, where $J$ is the
nearest-neighbor exchange constant, while `nearly-critical' quantum-disordered
ground states have a energy-gap, $\Delta$, towards excitations with spin-1,
which satisfies $\Delta \ll J$. Under these circumstances, we show that the
wavevector/frequency-dependent uniform and staggered spin susceptibilities, and
the specific heat, are completely universal functions of just three
thermodynamic parameters. Explicit results for the universal scaling functions
are obtained by a $1/N$ expansion on the $O(N)$ quantum non-linear sigma model,
and by Monte Carlo simulations. These calculations lead to a variety of
testable predictions for neutron scattering, NMR, and magnetization
measurements. Our results are in good agreement with a number of numerical
simulations and experiments on undoped and lightly-doped $La_{2-\delta}
Sr_{\delta}Cu O_4$.
