This paper investigates whether structural breaks and long memory are relevant features in modeling and forecasting the conditional volatility of oil spot and futures prices using three GARCH-type models, i.e., linear GARCH, GARCH with structural breaks and FIGARCH. By relying on a modified version of Inclan and Tiao (1994)'s iterated cumulative sum of squares (ICSS) algorithm, our results can be summarized as follows. First, we provide evidence of parameter instability in five out of twelve GARCH-based conditional volatility processes for energy prices. Second, long memory is effectively present in all the series considered and a FIGARCH model seems to better fit the data, but the degree of volatility persistence diminishes significantly after adjusting for structural breaks. Finally, the out-of-sample analysis shows that forecasting models accommodating for structural break characteristics of the data often outperform the commonly used short-memory linear volatility models. It is however worth noting that the long memory evidence found in the in-sample period is not strongly supported by the out-of-sample forecasting exercise.