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The problem of stabilizing reference trajectories - also referred to as the trajectory tracking problem - for nonholonomic mobile robots is revisited. Theoretical difficulties and impossibilities that set inevitable limits to what is achievable with feedback control are surveyed, and properties of kinematic control models are recalled, with a focus on controllable driftless systems that are invariant on a Lie group. This geometric framework takes advantage of ubiquitous symmetry properties involved in the motion of mechanical bodies. The transverse function approach, a control design method developed by the authors for the past few years, is reviewed. A salient feature of this approach, which singles it out of the abundant literature devoted to the subject, is the obtention of feedback laws that unconditionally achieve the practical stabilization of arbitrary reference trajectories, including fixed points and nonadmissible trajectories. This property is complemented with novel results showing how the more common property of asymptotic stabilization of a large class of admissible trajectories can also be granted with this type of control. Application to unicycle-type and car-like vehicles is presented and illustrated via simulations. Complementary issues (transient maneuvers monitoring, extensions of the approach to systems that are not invariant on a Lie group, etc.) are also addressed with the concern of practicality.