Compactifications of string theory on Calabi-Yau threefolds lead to supersymmetric four-dimensional vacua with unstable moduli. In order to stabilise these moduli, one may introduce background fluxes in the compact 6-manifold. However, such fluxes backreact on the internal geometry so that the Calabi-Yau condition is broken and a weaker condition of reduced structure group is imposed instead. In contrast to the vast number of Calabi-Yau manifolds, few manifolds with the relevant structure group have been constructed, and this lack of examples has left important properties of flux compactifications in obscurity. In this talk, I will report on recent progress in the construction of SU(3) structures on 6-dimensional smooth compact toric varieties (SCTVs). I will review the topological criterium for the existence of an SU(3) structure on a 6-manifold, which can be fulfilled on an infinite class of SCTVs. Since in string vacua the torsion of the SU(3) structure are constrained, I will then present a method to explicitly construct the SU(3) structure and compute its torsion. I will discuss when parametric choices can be made to tune the torsion classes, and illustrate the construction with several examples.