State-alternating context-free grammars are introduced, and the language classes obtained from them are compared to the classes of the Chomsky hierarchy as well as to some well-known complexity classes. In particular, state-alternating context-free grammars are compared to alternating context-free grammars (Theoret. Comput. Sci. 67 (1989) 75-85) and to alternating pushdown automata. Further, various derivation strategies are considered, and their influence on the expressive power of (state-) alternating context-free grammars is investigated. (c) 2005 Elsevier B.V. All rights reserved.