Introduction -- Preliminaries -- Part I Theoretical foundation: Vanishing level value and convergence to zero-level set -- Path following control in 3d using a vector field -- Topological analysis of vector-field guided path following on manifolds -- The domain of attraction of the desired path in vector-field guided path following -- Refined dichotomy convergence in vector-field guided path-following on Rn -- Part II Applications with formal guarantees: Guiding vector fields for following occluded paths -- A singularity-free guiding vector field for robot navigation -- Guiding vector fields for multi-robot coordinated navigation -- Conclusions and future research -- III Appendix.