  We present the theory of nonzero temperature ($T$) spin dynamics and
transport in one-dimensional Heisenberg antiferromagnets with an energy gap
$\Delta$. For $T << \Delta$, we develop a semiclassical picture of thermally
excited particles. Multiple inelastic collisions between the particles are
crucial, and are described by a two-particle S-matrix which has a
super-universal form at low momenta. This is established by computations on the
O(3) $\sigma$-model, and strong and weak coupling expansions (the latter using
a Majorana fermion representation) for the two-leg S=1/2 Heisenberg
antiferromagnetic ladder. As an aside, we note that the strong-coupling
calculation reveals a S=1, two particle bound state which leads to the presence
of a second peak in the T=0 inelastic neutron scattering (INS) cross-section
for a range of values of momentum transfer. We obtain exact, or numerically
exact, universal expressions for the thermal broadening of the quasi-particle
peak in the INS cross-section, for the magnetization transport, and for the
field dependence of the NMR relaxation rate $1/T_1$ of the effective
semiclassical model: these are expected to be asymptotically exact for the
quantum antiferromagnets. The results for $1/T_1$ are compared with the
experimental findings of Takigawa et al and the agreement is quite good. In the
regime $\Delta < T < (a typical microscopic exchange)$ we argue that a
complementary description in terms of semiclassical waves applies, and give
some exact results for the thermodynamics and dynamics.
