Abstract. We determine the holomorphic mappings of ${\msbm C}^n$ that induce bounded composition operators on the Fock space in ${\msbm C}^n$. Furthermore, we determine which of these composition operators are compact, and we compute the operator norm of all bounded composition operators in this setting. We also consider extensions of these results to various generalizations of the Fock space.

AMS Subject Classification (1991): 47B33; 32A37

Received May 28, 2002, and in revised form November 15, 2002. (Registered under 2937/2009.)