Over the past twenty years, a substantial amount of research has been performed to expand the understanding and modelling capabilities of energy geostructures: innovative earth-contact structures that provide combined structural support and renewable energy supply. The numerical modelling of energy geostructures has been progressively improved in capabilities and effectiveness, representing a critical solution for any comprehensive analysis and design of such geostructures. However, despite the observed rise in computational capabilities, the numerical modelling of energy geostructures remains daunting, especially for the engineering practice. In this context, analytical solutions based on sound theoretical principles and validated through experimental evidence represent paramount tools for the analysis and design of energy geostructures. This paper summarises the only analytical solutions currently available for the analytical modelling of a key problem in the analysis and design of all energy geostructures: the response of such geostructures to the thermal and mechanical actions that are applied as a consequence of their structural support and energy supply. Cylindrical and plane energy geostructures, such as energy piles and walls (or slabs) are considered, respectively.