  We study the dynamics of the East model, comprising a chain of uncoupled
spins in a downward-pointing field. Glassy effects arise at low temperatures
$T$ from the kinetic constraint that spins can only flip if their left
neighbour is up. We give details of our previous solution of the
non-equilibrium coarsening dynamics after a quench to low $T$ (Phys. Rev.\
Lett. 83:3238, 1999), including the anomalous coarsening of down-spin domains
with typical size $\bar{d} \sim t^{T \ln 2}$, and the pronounced `fragile
glass'-divergence of equilibration times as $t_*=\exp(1/T^2\ln 2)$. We also
link the model to the paste-all coarsening model, defining a family of
interpolating models that all have the same scaling distribution of domain
sizes. We then proceed to the problem of equilibrium dynamics at low $T$. Based
on a scaling hypothesis for the relation between timescales and lengthscales,
we propose a model for the dynamics of `superdomains' which are bounded by
up-spins that are frozen on long timescales. From this we deduce that the
equilibrium spin correlation and persistence functions should exhibit identical
scaling behaviour for low $T$, decaying as $g(\tilde{t})$. The scaling variable
is $\tilde{t}=(t/t_*)^{T\ln 2}$, giving strongly stretched behaviour for low
$T$. The scaling function $g(\cdot)$ decays faster than exponential, however,
and in the limit $T\to 0$ at fixed $\tilde{t}$ reaches zero at a {\em finite}
value of $\tilde{t}$.
