This paper investigates the volatility spillovers and co-movements among oil prices and stock prices of major oil and gas corporations over the period between 18th June 2001 and 1st February 2016. To do so, we use the spillover index approach by Diebold and Yilmaz (2009, 2012, 2014, 2015) and the dynamic correlation coefficient model of Engle (2002) so as to identify the transmission mechanisms of volatility shocks and the contagion of volatility among oil prices and stock prices of oil and gas companies, respectively. Given that volatility transmission across oil and major oil and gas corporations is important for portfolio diversification and risk management, we also examine optimal weights and hedge ratios among the aforementioned series. Our results point to the existence of significant volatility spillover effects among oil and oil and gas companies' stock volatility. However, the spillover is usually unidirectional from oil and gas companies' stock volatility to oil volatility, with BP, CHEVRON, EXXON, SHELL and TOTAL being the major net transmitters of volatility to oil markets. Conditional correlations are positive and time-varying, with those between each of the aforementioned companies and oil being the highest. Finally, the diversification benefits and hedging effectiveness based on our results are discussed. (C) 2018 Elsevier B.V. All rights reserved.