The purpose of this talk is to discuss some positivity questions in hyper-Kähler geometry. The first part will be a short introduction to positivity in algebraic geometry, with an emphasis on the themes I will focus on during the second part. To be more precise, given an irreducible holomorphic symplectic (IHS) manifold, we study the volume function, the structure of the big cone and of the pseudo-effective cone and, time permitting, some convex bodies associated with any big divisor.