© 2013 Dr. Jose Manuel Ayala Hoffmann ; A bounded curvature path corresponds to a C¹ and piecewise C² path lying in R² having its curvature bounded by a positive constant and connecting two elements of the tangent bundle TR². In this thesis we develop techniques to answer questions about the connectivity of the spaces of bounded curvature paths and to establish the minimal length elements in spaces bounded curvature paths. Let Γ(n) be the space of bounded curvature paths having fixed initial and final points x,y in TR² and having winding number n. This thesis concentrates on the following problems: • The classification of the minimal length elements in Γ(n) for all n and all x,y in TR². • The classification of the homotopy classes in Γ(n) for all n and all x,y in TR².