Given a smooth family P of complex univariate polynomials, it is natural to study the regularity of its roots. I shall give an overview of the known results and recent developments. In particular, I will show that the roots of a quasianalytic multiparameter family P can be chosen smoothly after applying finitely many local blow-ups and local power substitutions. Using that, I will prove that the roots of P admit a parameterization by functions of bounded variation (not better!), locally. Similar results can be obtained for the eigenvalues and eigenvectors of a quasianalytic multiparameter family of normal matrices.