  We associate to a finite digraph $D$ a lattice polytope $P_D$ whose vertices
are the rows of the Laplacian matrix of $D$. This generalizes a construction
introduced by Braun and the third author. As a consequence of the Matrix-Tree
Theorem, we show that the normalized volume of $P_D$ equals the complexity of
$D$, and $P_D$ contains the origin in its relative interior if and only if $D$
is strongly connected. Interesting connections with other families of simplices
are established and then used to describe reflexivity, $h^*$-polynomial, and
integer decomposition property of $P_D$ in these cases. We extend Braun and
Meyer's study of cycles by considering cycle digraphs. In this setting we
characterize reflexivity and show there are only four non-trivial reflexive
Laplacian simplices having the integer decomposition property.
