A frequent dilemma in the design of a database programming language is the choice between a language with a rich set of tailored notations for schema definitions, query expressions, etc., and a small, simple core language. We address this dilemma by proposing extensible grammars, a syntax-definition formalism for incremental language extensions and restrictions based on an initial core language. The translation of programs written in rich object languages into a small core language is defined via syntax-directed patterns. In contrast to macro-expansion and program-rewriting tools, our extensible grammars respect scoping rules. Therefore, we can introduce binding constructs while avoiding problems with unwanted name clashes. We develop extensible grammars and illustrate their use by extending the lambda calculus with let-bindings, conditionals, and SQL-style query expressions. We then give a formal description of the underlying parsing, transformat,ion, and substitution rules. Finally,...