We study the positivity and regularity of solutions to the fractional porous medium equations ut+(-)sum=0 in (0,)x for m > 1 and s (0,1), with Dirichlet boundary data u = 0 in (0,)x(N\) and nonnegative initial condition u(0,)=u00. Our first result is a quantitative lower bound for solutions that holds for all positive times t > 0. As a consequence, we find a global Harnack principle stating that for any t > 0 solutions are comparable to d(s/m), where d is the distance to . This is in sharp contrast with the local case s = 1, where the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior (C in x and C-1,C- in t) and establish a sharp Cxs/m</mml:msubsup> regularity estimate up to the boundary. Our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the form <mml:msub>ut+LF(u)=0 in , both in bounded and unbounded domains.(c) 2016 Wiley Periodicals, Inc.