The paper studies the question of the existence of a model companion of a theory expanded by an automorphism. The authors obtain a complete characterization of all model-complete theories expanding the theory of non-trivial dense linear orders without endpoints in the language \(\{<,\sigma\}\) where \(\sigma\) is an increasing automorphism. They also obtain a complete characterization of all model-complete theories of omega-sums in the class of ordered abelian groups in the language \(\{+, -, 0, <,\sigma\}\) where \(\sigma\) is a “piecewise” increasing automorphism. At last, they show that the theory of real closed fields does not have a model companion in the language \(\{+, -, \times, 0, 1, <, \sigma\}\) where \(\sigma\) is an increasing automorphism.