We present parsing algorithms for various mildly non-projective dependency formalisms. In particular, algorithms are presented for: all well-nested structures of gap degree at most 1, with the same complexity as the best existing parsers for constituency formalisms of equivalent generative power; all well-nested structures with gap degree bounded by any constant k; and a new class of structures with gap degree up to k that includes some ill-nested structures. The third case includes all the gap degree k structures in a number of dependency treebanks.