Digitalisation of the process and energy industries through energy digital twin technology promises step-improvements in energy management and optimisation, better servicing and maintenance, energy-efficient design and evolution of existing sites, and integration with locally and regionally generated renewable energy. This systematic and critical review aims to accelerate the understanding, classification, and application of energy digital twin technology. It adds to the literature by developing an original multi-dimensional digital twin classification framework, summarising the applications of energy digital twins throughout a site's lifecycle, and constructing a proposal of how to apply the technology to industrial sites and local areas to enable a reduction in carbon and other environmental footprints. The review concludes by identifying key challenges that face uptake of energy digital twins and a framework to apply the energy digital twins.