Cycles of given length in oriented graphs

We show that for each ℓ⩾4 every sufficiently large oriented graph G with δ+(G),δ−(G)⩾⌊|G|/3⌋+1 contains an ℓ-cycle. This is best possible for all those ℓ⩾4 which are not divisible by 3. Surprisingly, for some other values of ℓ, an ℓ-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an ℓ-cycle (with ℓ⩾4 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity and consider ℓ-cycles in general digraphs.