We present a new apl)roach , ilhlstrated by two algorithms> for parsing not only Finite SI.ate (:Iranlnlars but also Context Free Grainlnars and their extension, by means of finite state machines. '/'he basis is the computation of a flxed point of a linite-state function, i.e. a finite-state transducer. Using these techniques, we have built a program that parses French sentences with a gramnlar of more than 200>000 lexical rules with a typical response time of less than a second. The tirst algorithm computes a fixed point of a non-deterluinistic tinite-state transducer and the second coniplites a lixed point of a deterministic bidirectiollal device called a bimachine. These two algoril;hms point out a new connection between the theory of parsing and the theory of representation of rational transduetions.