  This paper deals with the large time dynamics of bounded solutions of
reaction-diffusion equations with unbounded initial support in $\mathbb{R}^N$.
We prove a variational formula for the spreading speeds in any direction, and
we also describe the asymptotic shape of the level sets of the solutions at
large time. The Freidlin-G\"artner type formula for the spreading speeds
involves newly introduced notions of bounded and unbounded directions of the
initial support. The results hold for a large class of reaction terms 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 also prove some
results of independent interest on some conditions guaranteeing the spreading
of solutions with large initial support and the link between these conditions
and the existence of traveling fronts with positive speed. The proofs use a mix
of ODE and PDE methods, as well as some geometric arguments. The results are
sharp and counterexamples are shown when the assumptions are not all fulfilled.
