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Abstract. We survey some recent progress in the theory of dynamical zeta-functions and explain its implications for counting problems. 0. The results One particularly elegant aspect of dynamical zeta functions, particularly in the context of hyperbolic ows, is the analogy with the Riemann zeta function in num-ber theory. Following dierent earlier denitions by Selberg and Smale[1967], Ruelle [1976b] proposed a formal denition of a dynamical zeta function for such a ow of the following form (s)