Khoussainov and Nerode [14] posed various open questions on model-theoretic properties of automatic structures. In this work we answer some of these questions by showing the following results: There is an uncountably categorical but not countably categorical theory for which only the prime model is automatic; There are complete theories with exactly 3,4,5,…3,4,5,… countable models, respectively, and every countable model is automatic; There is a complete theory for which exactly 2 models have an automatic presentation; If LOGSPACE=PLOGSPACE=P then there is (...) an uncountably categorical but not countably categorical theory for which the prime model does not have an automatic presentation but all the other countable models are automatic; There is a complete theory with countably many countable models for which the saturated model has an automatic presentation but the prime model does not have one. (shrink)