In this paper, we analyze daily data-based price transmission and volatility spillovers between crude oil and bond markets of major oil exporters and importers, by accounting for structural shifts as a smooth process in causality and volatility spillover estimations. In general, we find that, oil prices tend to predict bond prices in majority of oil exporting countries, and for the two major oil importers of India and China. But, the feedback from bond to oil prices is weak, and is detected for China and USA. Regarding volatility spillovers, oil volatility affects the bond market volatility of some major oil exporters (Kuwait, Norway and Russia), and an importer (France). However, the most prominent volatility spillovers are from bond to oil, except for Kuwait and Saudi Arabia. We also reveal that taking into account for smooth structural shifts - accounting for structural breaks - strengthens our findings and particularly is important for volatility spillover analysis. Our results have important implications for academics, investors, and policy makers.