A bounded linear operator T on a complex Hilbert space will be called completely indecomposable if its spectrum is not a singleton, and is included in the spectrum of the restrictions of T and T* to any of their nonzero invariant subspaces. Two classes of completely indecomposable operators are constructed. The first consists of essentially selfadjoint operators with spectrum [-2, 2], and the second of bilateral weighted shifts whose spectrum is the unit circle. We do not know whether any of the operators in the first class has a proper invariant subspace and if any of the operators in the second class has a proper hyperinvariant subspace. We also establish a new uniqueness theorem of Cartwright-Levinson type which is the main ingredient in our proofs of complete indecomposability.