This dissertation presents a new algorithm for parsing left-corner context-free grammars and develops extensions of context-free parsing concepts to indexed grammars. It presents a complete definition and algorithms for a class of indexed LL-parsable (ILL) grammars. These parsers are based on two-level pushdown automata. It also reports some of the problems in trying to extend left-corner techniques to indexed grammars. It includes an extensive bibliography of extensions to context-free grammars.