Bell polynomials appear in several combinatorial constructions throughout
mathematics. Perhaps most naturally in the combinatorics of set partitions, but
also when studying compositions of diffeomorphisms on vector spaces and
manifolds, and in the study of cumulants and moments in probability theory. We
construct commutative and noncommutative Bell polynomials and explain how they
give rise to Fa\`a di Bruno Hopf algebras. We use the language of incidence
Hopf algebras, and along the way provide a new description of antipodes in
noncommutative incidence Hopf algebras, involving quasideterminants. We also
discuss M\"obius inversion in certain Hopf algebras built from Bell
polynomials.Comment: 37 pages, final version, to appear in IJA
