  We study the question of positivity of the fundamental solution for
fractional diffusion and wave equations of the form, which may be of fractional
order both in space and time. We give a complete characterization for the
positivity of the fundamental solution in terms of the order of the time
derivative $\alpha\in(0,2)$, the order of the spatial derivative $\beta\in
(0,2]$ and the spatial dimension $d$. It turns out that the fundamental
solution fails to be positive for all $\alpha\in (1,2)$, and either $\beta\in
(0,2]$ and $d\ge 2$ or $\beta<\alpha$ and $d=1$, whereas in the other cases it
remains positive. The proof is based on delicate properties of the Fox
H-functions and the Mittag-Leffler functions.
