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 $\mathcal{S}(\mathbf{k},\omega ),$ the 2D magnetic correlation length $\xi \mathrm{}(x,T)\mathrm{}$ and, for the quasi- (2D) case, the Néel temperature $T_N\mathrm{}\left(x\right).$ We find that in two dimensions a hydrodynamic description of excitations in terms of spin waves breaks down at wavelengths larger than $l/a\sim e^{\pi /4x},$ 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 $a^{-1}\gg k\gg l^{-1}$ consists of two distinct parts: (i) a damped quasiparticle peak at an energy $c_{0}k\gtrsim \omega \gg {\omega }_0,$ with abnormal damping ${\Gamma }_k\sim x{c}_{0}k,$ where ${\omega }_0\sim c_0{l}^{-1},$ $c_0$ is the bare spin-wave velocity; and (ii) a non-Lorentian localization peak at $\omega \sim {\omega }_0.$ For $k\lesssim l^{-1}$ these two structures merge, and the spectrum becomes incoherent. The density of states acquires a constant term, and exhibits an anomalous peak at $\omega \sim {\omega }_0$ 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 $D=2\mathrm{}$ is not the lower critical dimension for this problem. We find that at small x the average staggered magnetization at the magnetic site is $M(x,0\mathrm{})\mathrm{}\simeq S-\Delta -Bx,$ where $\Delta$ is the zero-point spin deviation and $B\simeq 0.21$ is independent of the value of S; the Néel temperature $T_N\mathrm{}\left(x\right)\mathrm{}\simeq \left(1\mathrm{}-A_{s}x\right)\hspace{0.5em}{T}_N\mathrm{}\left(0\right),$ where $A_s=\pi -2/\pi +B/\left(S-\Delta \right)$ 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 $x_p\approx \mathrm{0.41.}$ This result suggests that long-range order is stable at small x, and can be lost only around $x\simeq x_p$ where approximations of our approach become invalid.