In this study we examine the dynamic structural relationship between oil price shocks and stock market returns and volatility for a sample of both net oil-exporting and net oil-importing countries between 1995:09 and 2013:07. We accomplish that, by extending the Diebold and Yilmaz (2012) dynamic spillover index using structural forecast error variance decomposition. The results for both stock market returns and volatility suggest that spillover effects vary across different time periods, and that this time-varying character is aligned with certain developments that take place in the global economy. In particular, aggregate demand shocks appear to act as the main transmitters of spillover effects to stock markets during periods characterised by economic-driven events, while supply-side and oil-specific demand shocks during periods of geopolitical unrest. Furthermore, differences regarding the directions and the strength of spillover effects can be reported both between and within the net oil-importing and net oil-exporting countries. These results are of particular importance to investors and portfolio managers, given the recent financialisation of the oil market.