If you ask a specialist in holomorphically symplectic varieties what they are, and why these objects are interesting, you can get very different answers, depending on whether that person comes from algebraic geometry, or from differential geometry. Or maybe they come from complex analysis, theoretical physics, representation theory, or number theory. In all these areas holomorphically symplectic manifolds play their own exceptional role, and in order to study them one needs to combine all these different points of view. In my talk, I will discuss the main properties of holomorphically symplectic manifolds: some of them follow immediately from the definition, and some are deep and difficult theorems. In addition, I will try to explain why it is so important to construct new examples of holomorphically symplectic manifolds, and why this problem is incredibly difficult.