In recent years, there has been growing interest in the market interactions between carbon (or clean/renewable energy) and traditional fossil energy such as coal and oil, but few studies have discussed their dynamic volatility spillover and time-varying correlation. To investigate these issues, we used the weekly data of the European Union carbon emission allowance (EUA) futures, biofuel and Brent oil prices from 25 October 2009 to 5 July 2020. We employed the vector autoregressive-generalized autoregressive conditional heteroscedasticity (VAR-GARCH) model with the Baba, Engle, Kraft and Krone (BEKK) specification. Our main findings are summarized as follows: First, we identified the sudden changes and the volatility persistence in the EUA, biofuel, and Brent oil markets, and also confirmed that the volatility of the markets has changed significantly over time. Second, we found a weak volatility spillover effect among the three markets, and a strong spillover effect between the EUA and Brent oil markets. In particular, the effect of volatility spillover from the Brent oil market to the EUA market was the strongest than the other cases. Lastly, in financial market, by holding the EUA and energy sources together as assets, investors can effectively hedge their investment risk. The possibility of hedging is more pronounced between the EUA and biofuel markets.