  We introduce the Kodaira dimension of contact 3-manifolds and establish some
basic properties. In particular, contact 3-manifolds with distinct Kodaria
dimensions behave differently when it comes to the geography of various kinds
of fillings. On the other hand, we also prove that, given any contact
3-manifold, there is a lower bound of $2\chi+3\sigma$ for all its minimal
symplectic fillings. This is motivated by the bound of Stipsicz for Stein
fillings. Finally, we discuss various aspects of exact self cobordisms of
fillable 3-manifolds.
