Every pseudo-Anosov mapping class $\phi$ deﬁnes an associated veering triangulation $\tau_\phi$ of a punctured mapping torus. In joint work with S.J. Taylor and D. Futer, we show that generically, $\tau_\phi$ is not geometric. After defining the objects of interest, we will focus on an important lemma: for every surface $\Sigma$, there is a pseudo-anosov map on $\Sigma$ whose associated veering triangulation is non-geometric. Establishing this lemma boils down to finding mapping tori that fiber over many surfaces, and our approach will be to find examples for which we can describe the fibered faces of the Thurston norm unit ball. Although we do this by hand in this case, we will also briefly discuss recent work with S. Tillmann and D. Cooper that gives an algorithmic approach for computing the norm ball.