  Non-equilibrium dynamics of classical random Ising spin chains are studied
using asymptotically exact real space renormalization group. Specifically the
random field Ising model with and without an applied field (and the Ising spin
glass (SG) in a field), in the universal regime of a large Imry Ma length so
that coarsening of domains after a quench occurs over large scales. Two types
of domain walls diffuse in opposite Sinai random potentials and mutually
annihilate. The domain walls converge rapidly to a set of system-specific
time-dependent positions {\it independent of the initial conditions}. We obtain
the time dependent energy, magnetization and domain size distribution
(statistically independent). The equilibrium limits agree with known exact
results. We obtain exact scaling forms for two-point equal time correlation and
two-time autocorrelations. We also compute the persistence properties of a
single spin, of local magnetization, and of domains. The analogous quantities
for the spin glass are obtained. We compute the two-point two-time correlation
which can be measured by experiments on spin-glass like systems. Thermal
fluctuations are found to be dominated by rare events; all moments of truncated
correlations are computed. The response to a small field applied after waiting
time $t_w$, as measured in aging experiments, and the fluctuation-dissipation
ratio $X(t,t_w)$ are computed. For $(t-t_w) \sim t_w^{\hat{\alpha}}$,
$\hat{\alpha} <1$, it equals its equilibrium value X=1, though time
translational invariance fails. It exhibits for $t-t_w \sim t_w$ aging regime
with non-trivial $X=X(t/t_w) \neq 1$, different from mean field.
