I will speak about the structure of cohomology of hyperkahler manifolds, and as this topic attracts huge attention and has a lot of thing to discuss, I will mainly focus on the bounds for b_2. I will start with a short introduction on the main tools involved: LLV-decomposition, Rozansky-Witten invariants. And then I will speak on the older result of Verbitsky-Soldatenkov and myself on the generalized Kuga-Satake construction, connection to Nagai conjecture, and recent progress made by Green-Kim-Laza-Robles, Sawon and myself.