In the context of energy price modelling, prices are formed from exponential maps of underlying factor processes, and the mathematical convenience this offers means that this is no surprise. In this talk we will show various ways in which models based on polynomial maps of polynomial processes (PMPP models) can function in a similar way. Polynomial processes have the property that expectations of polynomial functions of the future state of the process, conditional on the current state, are themselves polynomial functions of the current state. It is this property that means that PMPP models also provide a level of mathematical convenience (for forming futures prices). But they also provide an additional level of flexibility, which means that they are capable of capturing the extreme dynamics that are commonly seen in energy market prices even with relatively tame dynamics in the underlying factor process. We will end by discussing numerical methods for the valuation of energy contracts in the PMPP setting.