In this paper, we test for linear and nonlinear Granger causality between the French, German, Japanese, UK and US daily stock index returns from 1973 to 2003. We find a strong contemporaneous linear dependence between European countries and a directional linear dependence from the US towards the other markets. Besides, linear causality increases after 1987, a finding consistent with the expected effects of financial liberalization of the 1980s and the 1990s. Above all, we document the presence of bidirectional nonlinear causality between daily returns. To check for spurious nonlinear causality, we filter out heteroskedasticity using a FIGARCH model. The dramatic decrease in the number of significant nonlinear causality lags confirms that heteroskedasticity played a major part in the previous findings. We then check if a few structural breaks can explain the remaining nonlinear causality. We find that a large number of nonlinear relationships vanish when we control for structural breaks, whereas linear causality remains.