This is a bit off the top of my head. So if someone feels I omitted some important development, feel free to mention it to me in a comment, by email, or in your own answer. One can also look at Burger's Seminaire Bourbaki report on Kähler groups from 2010 for a more recent survey.
- A natural question, open in the 90's, is what one-relator Kähler groups look like. This was settled by Biswas-(Mahan)Mj and Kotschick. They are fundamental groups of (orbifold) curves.
- Delzant proved that solvable non-residually nilpotent groups aren't Kähler. (In the 90's Nori and I proved this under certain finiteness conditions, which were unnecessary in hindsight.)
- It's still an open problem to determine whether the class of Kähler groups is the same as the class of fundamental groups of smooth complex projective varieties. Campana-Claudon-Eyssidieux showed that for linear groups these classes are "virtually" the same.
- It's still open to decide whether mapping class groups are Kähler in general. For genus 2 and 3, they aren't (Veliche, Hain resp.)
- Infinite three manifold groups aren't Kähler (Dimca-Suciu).
