Invariant and non-invariant almost complex structures on compact quotients of Lie groups

In this article, we briefly survey the classical problem of understanding which Lie algebras admit a complex structure, put in the broader perspective of almost complex structures with special properties. We focus on the different behavior of invariant and non-invariant structures, with special attention to their canonical bundle and Kodaira dimension. We provide new examples of computations of the Kodaira dimension of invariant and non-invariant structures.