In this lecture, we address Serrin-type problems in Riemannian manifolds. We begin by establishing a Heintze-Karcher inequality and proving a Soap Bubble theorem, with its corresponding rigidity, in the context of ambient spaces with a Ricci tensor bounded from below. We then focus on Serrin’s problem within bounded domains of manifolds endowed with a conformal vector field. A key tool in this analysis is a new Pohozaev identity that incorporates the scalar curvature of the manifold. Applications involve Einstein and constant scalar curvature spaces. This lecture is based on joint works with A. Roncoroni (Politecnico di Milano, Italy), M. Santos (UFPB, Brazil), M. Andrade (UFS, Brazil), and Diego Marín (Universidad de Granada, Spain)