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We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I ⊆ 𝕜[x 0, … , xn ] we show that 
for all positive integers m, t and r, where e is the big-height of I and 
. This captures two conjectures (r = 1 and r = e): one of Harbourne and Huneke, and one of Bocci et al. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.
