The discriminant of a map f from the plane to the plane is the image of its singular set. In general, the discriminant is a curve in the target plane, which we assume is endowed with the Euclidean metric. Key local geometric features of the discriminant curve are its singularities, inflections, and vertices. A deformation of a singularity of f results in a deformation of its discriminant. Several questions naturally arise: Observe that diffeomorphisms cannot be applied in the target, as they destroy the geometry of the discriminant. Answers to the above questions are provided throughout the book for the so-called fold, cusp, swallowtail, lips/beaks, butterfly, goose, and gull singularities. This chapter defines the notion of geometric deformations of the discriminant of a map-germ from the plane to the plane, taking into account both the singularities and the geometry of the discriminant. An equivalence relation between families of map-germs that preserves both singularities and geometry is also introduced.