Abstract. A tensor T of rank k is identifiable when it has a unique decomposition in terms of rank-1 tensors. There are cases in which the identifiability fails over C, for general tensors of fixed rank. The failure, often, is due to the existence of an elliptic normal curve through general points of the corresponding variety of rank-1 tensors. After a brief introduction to the subject, we prove the existence of non-empty euclidean open subsets of some varieties of real k-rank tensors, whose elements have 2 complex decompositions, but are identifiable over R. Moreover we provide examples of non-trivial euclidean open subsets in certain spaces of symmetric tensors and of almost unbalanced tensors, whose elements have real rank equal to the complex rank and are identifiable over R but not over C. On the contrary, there are examples of tensors of given real rank, for which identifiability over R can't hold in non-trivial open subsets. These results have been obtained in collaboration with Cristiano Bocci and Luca Chiantini.