This paper examines the impact of structural oil price shocks on the covariance of U.S. stock market return and stock market volatility. We construct from daily data on return and volatility the covariance of return and volatility at monthly frequency. The measures of daily volatility are realized-volatility at high frequency (normalized squared return), conditional-volatility recovered from a stochastic volatility model, and implied-volatility deduced from options prices. Positive shocks to aggregate demand and to oil-market specific demand are associated with negative effects on the covariance of return and volatility. Oil supply disruptions are associated with positive effects on the covariance of return and volatility. The spillover index between the structural oil price shocks and covariance of stock return and volatility is large and highly statistically significant.