The "local lifting problem" asks: given a finite group G and an embedding i: G --> (continuous k-automorphisms of k[[t]]), where k is algebraically closed of characteristic p, does there exist a finite extension R/W(k) and a map j: G --> (continuous R-automorphisms of R[[t]]) such that j lifts i? (Here W(k) is the ring of Witt vectors of k.) The Oort conjecture states that the local lifting problem should always have a solution for G cyclic. Equivalently, it says that every cyclic Galois cover of Spec k[[x]] lifts to characteristic zero. This is basic Kummer theory when p does not divide G, and has been proven when v_p(|G|) = 1 (Oort, Sekiguchi, Suwa) and when v_p(|G|) = 2 (Green, Matignon). We will first motivate the local lifting problem from geometry, and then we will show that it has a solution for a large family of cyclic extensions. This family includes all extensions where v_p(|G|) = 3 and many extensions where v_p(|G|) is arbitrarily high. This is joint work with Stefan Wewers.