  We introduce an axiomatic definition for the Kodaira dimension and classify
Thurston geometries in dimensions $\leq 5$ in terms of this Kodaira dimension.
We show that the Kodaira dimension is monotone with respect to the partial
order defined by maps of non-zero degree between 5-manifolds. We study the
compatibility of our definition with traditional notions of Kodaira dimension,
especially the highest possible Kodaira dimension. To this end, we establish a
connection between the simplicial volume and the holomorphic Kodaira dimension,
which in particular implies that any smooth K\"ahler 3-fold with non-vanishing
simplicial volume has top holomorphic Kodaira dimension.
