Fundamental groups of rationally connected varieties. Tame coverings and fundamental groups of algebraic varieties. I: Branch loci with normal crossings; applications: Theorems of Zariski and Picard. II: Branch curves with higher singularities. III: Some other sets of conditions for the fundamental group to be abelian. IV: Product theorems. V: Three cuspidal plane quartics. VI: Plane curves of order at most four 