  We present a comprehensive study of the dynamical properties of the quantum
Heisenberg antiferromagnet on a triangular lattice within the framework of
spin-wave theory. The distinct features of spin-wave excitations in the
triangular-lattice antiferromagnet are (i) finite lifetime at zero temperature
due to spontaneous two-magnon decays, (ii) strong renormalization of magnon
energies with respect to the harmonic result, and (iii) logarithmic
singularities in the decay rate Gamma_k. Quantum corrections to the magnon
spectrum are obtained using both the on-shell and off-shell solutions of the
Dyson equation with the lowest-order magnon self-energy. At low-energies magnon
excitations remain well-defined albeit with the anomalous decay rate Gamma_k ~
k^2 at k->0 and Gamma_k ~ |k-Q_AF|^7/2 at k->Q_AF. At high energies, magnons
are heavily damped with the decay rate reaching (2 Gamma_k/E_k) ~ 0.3 for the
case S=1/2. The on-shell solution shows logarithmic singularities in Gamma_k
with the concomitant jump-like discontinuities in E_k along certain contours in
the momentum space. Such singularities are even more prominent in the magnon
spectral function A(k,w). Although the off-shell solution removes such
log-singularities, the decay rates remain strongly enhanced. We also discuss
the role of higher-order corrections and show that such singularities may lead
to complete disappearance of the spectrum in the vicinity of certain k-points.
The kinematic conditions for two-magnon decays are analyzed for various
generalizations of the triangular-lattice antiferromagnet as well as for the
XXZ model on a kagome lattice. Our results suggest that decays and
singularities in the spin-wave spectra must be ubiquitous in all these systems.
