Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)A classical result of Birkhoff states that every nontrivial translation operator on the space H (C) of entire functions of one complex variable is hypercyclic. Godefroy and Shapiro extended this result considerably by proving that every nontrivial convolution operator on the space H (C-n) of entire functions of several complex variables is hypercyclic. In sharp contrast with these classical results we show that no convolution operator on the space H (C-N) of entire functions of infinitely many complex variables is hypercyclic. On the positive side we obtain hypercyclicity results for convolution operators on spaces of entire functions on important locally convex spaces.761141158FAPESP [2014/50536-7]FAPEMIG [PPM-00086-14]CNPq [482515/2013-9, 307517/2014-4]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)