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We study the positivity and regularity of solutions to the fractional porousmedium equations $u_t+(-\Delta)^su^m=0$ in $(0,\infty)\times\Omega$, for $m>1$and $s\in (0,1)$ and with Dirichlet boundary data $u=0$ in$(0,\infty)\times({\mathbb R}^N\setminus\Omega)$, and nonnegative initialcondition $u(0,\cdot)=u_0\geq0$.  Our first result is a quantitative lower bound for solutions which holds forall positive times $t>0$. As a consequence, we find a global Harnack principlestating that for any $t>0$ solutions are comparable to $d^{s/m}$, where $d$ isthe distance to $\partial\Omega$. This is in sharp contrast with the local case$s=1$, in which the equation has finite speed of propagation.  After this, we study the regularity of solutions. We prove that solutions areclassical in the interior ($C^\infty$ in $x$ and $C^{1,\alpha}$ in $t$) andestablish a sharp $C^{s/m}_x$ regularity estimate up to the boundary.  Our methods are quite general, and can be applied to a wider class ofnonlocal parabolic equations of the form $u_t-\mathcal L F(u)=0$ in $\Omega$,both in bounded or unbounded domains.

Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains