The well-known Alexandrov theorem states that the only closed embedded surfaces with constant mean curvature in Euclidean space are the round spheres. There are many generalizations, commonly known as rigidity theorems. In this talk, I am going to illustrate how we can use the weighted Hsiung-Minkowski formulas to obtain simple proofs of these kinds of rigidity results. More precisely, I will give Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of warped product manifolds, including space forms. I will also show the rigidity of closed immersed self-expanding solitons to the weighted generalized inverse curvature flow. If time permits, I will illustrate other applications of the formulas. Part of this talk is a joint work with Hojoo Lee and Juncheol Pyo.