  We study reaction-diffusion equations of various types in the half-space. For
bistable reactions with Dirichlet boundary conditions, we prove conditional
uniqueness: there is a unique nonzero bounded steady state which exceeds the
bistable threshold on large balls. Moreover, solutions starting from
sufficiently large initial data converge to this steady state as $t \to
\infty$. For compactly supported initial data, the asymptotic speed of this
propagation agrees with the unique speed $c_*$ of the one-dimensional traveling
wave. We furthermore construct a traveling wave in the half-plane of speed
$c_*$.
  In parallel, we show analogous results for ignition reactions under both
Dirichlet and Robin boundary conditions. Using our ignition construction, we
obtain stronger results for monostable reactions with the same boundary
conditions. For such reactions, we show in general that there is a unique
nonzero bounded steady state. Furthermore, monostable reactions exhibit the
hair-trigger effect: every solution with nontrivial initial data converges to
this steady state as $t \to \infty$. Given compactly supported initial data,
this disturbance propagates at a speed $c_*$ equal to the minimal speed of
one-dimensional traveling waves. We also construct monostable traveling waves
in the Dirichlet or Robin half-plane with any speed $c \geq c_*$.
