In which ways can a given smooth manifold be curved in a specific manner? This question is of fundamental interest in Riemannian geometry and leads to the consideration of the space of all Riemannian metrics on this manifold which satisfy the desired curvature condition. The moduli space is then obtained by identifying Riemannian metrics which are isometric and the goal is to understand the topology of these spaces. In these talks I will give an overview on the situation if one considers Riemannian metrics with positive scalar curvature on closed manifolds whose dimension is at least 5. For that I will introduce the required tools from spin geometry and surgery theory.