Advances on Superelliptic Curves and Their Applications -- Preface -- Contents -- The case for superelliptic curves -- Weierstrass points of superelliptic curves -- Theta functions of superelliptic curves -- Cyclic curves over the reals -- Reduction theory of binary forms -- Sato-Tate groups of genus 2 curves -- Heights on algebraic curves -- Descent and Covering Collections -- Chabauty and theMordell-Weil Sieve -- Rational points on Jacobians of hyperelliptic curves -- Explicit p-adic methods for elliptic and hyperelliptic curves -- Error Correcting Quantum Codes and Algebraic Curves -- Graph based cubical multivariate maps and their cryptographical applications -- Weight distributions, zeta functions and Riemann hypothesis for linear and algebraic geometry codes -- Galois geometries, codes, and new invariants for incidence structures -- Subject Index -- Author Index.