Abstract: In 2010 Alesker and Verbitsky posed a conjecture on certain hyperhermitian manifolds in the same spirit of the famous Calabi conjecture solved by Yau in the Kähler setting. The so-called "quaternionic Calabi conjecture" is equivalent to a Monge-Ampère equation of quaternionic type. As of today the conjecture is still open in its full generality. We will briefly overview the relevant mathematical framework, explain the geometric significance of this problem, overview the current progress towards its solution and discuss other related elliptic and parabolic equations. This talk is based on joint works with Lucio Bedulli, Luigi Vezzoni and Jiaogen Zhang.