  We study the existence and properties of solutions and free boundaries of the
one-phase Stefan problem with fractional diffusion posed in $\mathbb{R}^N$. In
terms of the enthalpy $h(x,t)$, the evolution equation reads $\partial_t
h+(-\Delta)^s\Phi(h) =0$, while the temperature is defined as
$u:=\Phi(h):=\max\{h-L,0\}$ for some constant $L>0$ called the latent heat, and
$(-\Delta)^s$ stands for the fractional Laplacian with exponent $s\in(0,1)$.
  We prove the existence of a continuous and bounded selfsimilar solution of
the form $h(x,t)=H(x\,t^{-1/(2s)})$ which exhibits a free boundary at the
change-of-phase level $h(x,t)=L$. This level is located at the line (called the
free boundary) $x(t)=\xi_0 t^{1/(2s)}$ for some $\xi_0>0$. The construction is
done in 1D, and its extension to $N$-dimensional space is shown.
  We also provide well-posedness and basic properties of very weak solutions
for general bounded data $h_0$ in several dimensions. The temperatures $u$ of
these solutions are continuous functions that have finite speed of propagation,
with possible free boundaries. We obtain estimates on the growth in time of the
support of $u$ for solutions with compactly supported initial temperatures.
Besides, we show the property of conservation of positivity for $u$ so that the
support never recedes. On the contrary, the enthalpy $h$ has infinite speed of
propagation and we obtain precise estimates on the tail.
  The limits $L\to0^+$, $L\to +\infty$, $s\to0^+$ and $s\to 1^-$ are also
explored, and we find interesting connections with well-studied diffusion
problems. Finally, we propose convergent monotone finite-difference schemes and
include numerical experiments aimed at illustrating some of the obtained
theoretical results, as well as other interesting phenomena.
