Arnau Padrol is an expert in the combinatorial theory of polytopes. One of his research areas concerns realization spaces of polytopes, including universality theorems (in the sense of Mnëv) for neighborly and inscribed polytopes, as well as projectively unique polytopes. He has contributed to projects concerning the geometric realizability of combinatorial structures generalizing the associahedron. In particular, he has contributions in the study of polytopal realizations of the $$\nu$$-Tamari lattice, lattice quotients of the weak order, and $$g$$-vector fans of cluster algebras.