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.

 1. 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.
 2. 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.)
 3. 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. Campaman-Claudon- Eyssidieux showed that for linear groups these classes are "virtually" the same.
 4. 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.)
 5. Infinite three manifold groups aren't Kähler (Dimca-Suciu).