  We consider reaction-diffusion equations $\partial_tu=\Delta u+f(u)$ in the
whole space $\mathbb{R}^N$ and we are interested in the large-time dynamics of
solutions ranging in the interval $[0,1]$, with general unbounded initial
support. Under the hypothesis of the existence of a traveling front connecting
$0$ and $1$ with a positive speed, we discuss the existence of spreading speeds
and spreading sets, which describe the large-time global shape of the level
sets of the solutions. The spreading speed in any direction is expressed as a
Freidlin-G\"artner type formula. This formula holds under general assumptions
on the reaction and for solutions emanating from initial conditions with
general unbounded support, whereas most of earlier results were concerned with
more specific reactions and compactly supported or almost-planar initial
conditions. We then investigate the local properties of the level sets at large
time. Some flattening properties of the level sets of the solutions, if
initially supported on subgraphs, will be presented. We also investigate the
special case of asymptotically conical-shaped initial conditions. For
Fisher-KPP equations, we state some asymptotic local one-dimensional and
monotonicity symmetry properties for the elements of the $\Omega$-limit set of
the solutions, in the spirit of a conjecture of De Giorgi for stationary
solutions of Allen-Cahn equations. Lastly, we present some logarithmic-in-time
estimates of the lag of the position of the solutions with respect to that of a
planar front with minimal speed, for initial conditions which are supported on
subgraphs with logarithmic growth at infinity. Some related conjectures and
open problems are also listed.
