  We study the regularity properties of the solutions to the nonlinear equation
with fractional diffusion $$ \partial_tu+(-\Delta)^{\sigma/2}\varphi(u)=0, $$
posed for $x\in \mathbb{R}^N$, $t>0$, with $0<\sigma<2$, $N\ge1$. If the
nonlinearity satisfies some not very restrictive conditions: $\varphi\in
C^{1,\gamma}(\mathbb{R})$, $1+\gamma>\sigma$, and $\varphi'(u)>0$ for every
$u\in\mathbb{R}$, we prove that bounded weak solutions are classical solutions
for all positive times. We also explore sufficient conditions on the
non-linearity to obtain higher regularity for the solutions, even $C^\infty$
regularity. Degenerate and singular cases, including the power nonlinearity
$\varphi(u)=|u|^{m-1}u$, $m>0$, are also considered, and the existence of
classical solutions in the power case is proved.
