The Weyl energy on four-manifolds is a quadratic curvature functional related to the Chern-Gauss-Bonnet formula. It is conformally invariant, and its critical points are called Bach-flat metrics and provide an interesting generalization of Einstein metrics or self-dual and anti-self-dual metrics. We will show that, under suitable conditions, it is possible to decrease the total Weyl energy under connected sum between two manifolds. These gluing results might be particularly useful for understanding the compactness properties of Weyl-minimizing sequences of metrics. This is based on joint works with Andrea Malchiodi.