Summary: We consider stationary infinite moving average processes of the form \[ Y_n= \sum_{i=-\infty}^\infty c_i Z_{n+i}, \quad n\in \mathbb Z, \] where \((Z_i)_{i\in\mathbb Z}\) is a sequence of independent and identically distributed (i.i.d.) random variables with light tails and \((c_i)_{i\in\mathbb Z}\) is a sequence of positive and summable coefficients. By ‘light tails’ we mean that \(Z_0\) has a bounded density \(f(t)\sim \nu(t)\exp(-\psi(t))\), where \(\nu(t)\) behaves roughly like a constant as \(t\to\infty\) and \(\psi\) is strictly convex satisfying certain asymptotic regularity conditions. We show that the i.i.d. sequence associated with \(Y_0\) is in the maximum domain of attraction of the Gumbel distribution. Under additional regular variation conditions on \(\psi\), it is shown that the stationary sequence \((Y_n)_{n\in\mathbb N}\) has the same extremal behaviour as its associated i.i.d. sequence. This generalizes H. Rootzén’s results [Ann. Probab. 14, 612–652 (1986; Zbl 0604.60019); ibid. 15, 728–747 (1987; Zbl 0637.60025)] where \(f(t)\sim ct^\alpha \exp(-t^p)\) for \(c>0\), \(\alpha\in\mathbb R\) and \(p>1\).