We shall give a summary of some of the main ideas in the area, beginning with conics (that is, curves which can be described by linear and quadratic equations, such as circles, lines, etc) and the group law on an elliptic curve. We then move onto higher genus curves, describing the Jacobian of a curve and its group law, and points of finite order on the Jacobian. We describe how recent advances in the constructive side of the subject have allowed the solution of several specific problems concerning cycles of quadratic polynomials, and ℚ-derived polynomials (that is: polynomials all of whose roots are in ℚ, all of whose derivatives also have the same property).