*Abstract*: The Bakry-Émery Ricci tensor is a generalization of the classical Ricci tensor to the setting of weighted Riemannian manifolds, i.e. Riemannian manifolds whose Riemannian volume forms are weighted by a smooth function. While it was defined by Bakry and Émery in the context of diffusion processes, the Bakry-Émery Ricci tensor also appears naturally in other settings, such as Ricci flow, general relativity, and collapsed Ricci limit spaces. In analogy with important open problems in the Riemannian case, we consider the question of which manifolds admit a weighted Riemannian metric of positive Bakry-Émery Ricci curvature. We will show that surgery techniques that are used in the Riemannian case can be extended and improved in the weighted case. As application we will construct weighted Riemannian metrics of positive Bakry-Émery Ricci curvature on all closed, simply-connected spin 5-manifolds. This is joint work with Francesca Tripaldi.