What happens to the medial axis of a curve as this curve evolves through Mean Curvature Motion? Medial axes and curvature motions have applications in shape recognition and shape classification in computer vision; in this context, both relate closely to the distance transform of a curve. This thesis tries to explore as much as possible the properties of each of those three objects and the relations between them. The main results are:

     - A set of conditions on the validity of a medial axis transform (chapter 2);

     - A differential equation for the change of smooth parts of the medial axis when its generating curve evolves under Mean Curvature Motion (chapter 3);

     - A differential equation that describes the evolution of the distance transform of a curve under Mean Curvature Motion (chapter 5);

     - A classification of the possible generic changes in the medial axis of a curve evolving through Mean Curvature Motion (chapter 7).

     We also provide some pictures representing three-dimensional Mean Curvature Motion and three-dimensional Minimal Curvature Motion in the Appendix.

     Finally, for a slightly longer overview of the motivations behind these questions, we refer the reader to section 1.3 in this thesis.