The theme of this book is the interactions between group theory and algebra/geometry/number theory, showing ubiquity and power of the basic principle of Galois theory. The books presents recent developments in a major line of work about covers of the projective line (and other curves), their fields of definition and parameter spaces, and associated questions about arithmetic fundamental groups. This is intimately tied up with the Inverse Problem of Galois Theory, and uses methods of algebraic geometry, group theory and number theory.