In this paper the asymptotic behaviour of the Castelnuovo-Mumford regularity of powers of a homogeneous ideal I is studied. It is shown that there is a linear bound for the regularity of the powers I n whose slope is the maximum degree of a homogeneous generator of I , and that the regularity of I n is a linear function for large n. Similar results hold for the integral closures of the powers of I . On the other hand we give examples of ideals for which the regularity of the saturated powers is

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... asymptotically not a linear function, not even a linear function with periodic coefficients. Mathematics Subject Classifications (1991): 13C99, 13D45, 13F17.