We present some result on sub-Riemannian geodesics in Carnot groups recently obtained in collaboration with E. Le Donne, G.P. Leonardi, and R. Monti. A sub-Riemannian manifold is a manifold M endowed with a distinguished subbundle D of the tangent bundle and with a metric on D. A distance (called sub-Riemannian) on M can be defined on minimizing the length among curves which are tangent to D. One of the main open problems in the field is the regularity of length minimizers, that is not trivial due to the presence of the so called abnormal curves. We provide a characterization of abnormal geodesic in Carnot groups (i.e., certain Lie groups which are the infinitesimal models of sub-Riemannian manifolds) showing that they are contained in certain algebraic varieties; this poses several questions ranging from analysis to algebraic geometry. Applications to the problem of geodesics' regularity will be discussed.