  We investigate symbolic and regular powers of monomial ideals. For a
square-free monomial ideal $I$ in $k[x_0, \ldots, x_n]$ we show
$I^{t(m+e-1)-e+r)}$ is a subset of $M^{(t-1)(e-1)+r-1}(I^{(m)})^t$ for all
positive integers $m$, $t$ and $r$, where $e$ is the big-height of $I$ and $M =
(x_0, \ldots, x_n)$. This captures two conjectures ($r=1$ and $r=e$): one of
Harbourne-Huneke and one of Bocci-Cooper-Harbourne. We also introduce the
symbolic polyhedron of a monomial ideal and use this to explore symbolic powers
of non-square-free monomial ideals.
