The paper investigates some of the fundamental ideas of the context-free grammar theory, as they are applied to several extensions and subclasses of context-free grammars. For these grammar families, including multi-component grammars, tree-adjoining grammars, conjunctive grammars and Boolean grammars, a summary of the following properties is given: parse trees, language equations, closure under several operations, normal forms, parsing algorithms, representation in the FO(LFP) logic, representations by automata and by categorial grammars, homomorphic characterizations, hardest language theorems, pumping lemmata and other limitations, computational complexity.