We have studied the static and dynamic magnetic properties of two-dimensional (2D) and quasi-two-dimensional, spin-S, quantum Heisenberg antiferromagnets diluted with spinless vacancies. Using spin-wave theory and the T-matrix approximation we have calculated the staggered magnetization $M(x,T),$ the neutron scattering dynamical structure factor $\mathcal{S}(\mathbf{k},\ensuremath{\omega}),$ the 2D magnetic correlation length $\ensuremath{\xi}(x,T)$ and, for the quasi- (2D) case, the N\'eel temperature ${T}_{N}(x).$ We find that in two dimensions a hydrodynamic description of excitations in terms of spin waves breaks down at wavelengths larger than $l/a\ensuremath{\sim}{e}^{\ensuremath{\pi}/4x},$ x being the impurity concentration and a the lattice spacing. We find signatures of localization associated with the scale l, and interpret this scale as the localization length of magnons. The spectral function for momenta ${a}^{\ensuremath{-}1}\ensuremath{\gg}k\ensuremath{\gg}{l}^{\ensuremath{-}1}$ consists of two distinct parts: (i) a damped quasiparticle peak at an energy ${c}_{0}k\ensuremath{\gtrsim}\ensuremath{\omega}\ensuremath{\gg}{\ensuremath{\omega}}_{0},$ with abnormal damping ${\ensuremath{\Gamma}}_{k}\ensuremath{\sim}x{c}_{0}k,$ where ${\ensuremath{\omega}}_{0}\ensuremath{\sim}{c}_{0}{l}^{\ensuremath{-}1},$ ${c}_{0}$ is the bare spin-wave velocity; and (ii) a non-Lorentian localization peak at $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\omega}}_{0}.$ For $k\ensuremath{\lesssim}{l}^{\ensuremath{-}1}$ these two structures merge, and the spectrum becomes incoherent. The density of states acquires a constant term, and exhibits an anomalous peak at $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\omega}}_{0}$ associated with low-energy localized excitations. These anomalies lead to a substantial enhancement of the magnetic specific heat ${C}_{M}$ at low temperatures. Although the dynamical properties are significantly modified, we show that $D=2$ is not the lower critical dimension for this problem. We find that at small x the average staggered magnetization at the magnetic site is $M(x,0)\ensuremath{\simeq}S\ensuremath{-}\ensuremath{\Delta}\ensuremath{-}Bx,$ where $\ensuremath{\Delta}$ is the zero-point spin deviation and $B\ensuremath{\simeq}0.21$ is independent of the value of S; the N\'eel temperature ${T}_{N}(x)\ensuremath{\simeq}(1\ensuremath{-}{A}_{s}x) {T}_{N}(0),$ where ${A}_{s}=\ensuremath{\pi}\ensuremath{-}2/\ensuremath{\pi}+B/(S\ensuremath{-}\ensuremath{\Delta})$ is weakly S dependent. Our results are in quantitative agreement with recent Monte Carlo simulations and experimental data for $S=1/2,$ 1, and 5/2. In our approach long-range order persists up to a high concentration of impurities ${x}_{c}$ which is above the classical percolation threshold ${x}_{p}\ensuremath{\approx}0.41.$ This result suggests that long-range order is stable at small x, and can be lost only around $x\ensuremath{\simeq}{x}_{p}$ where approximations of our approach become invalid.