Indirect calorimetry (IC) is considered the gold standard for measuring resting energy expenditure (REE). This review presents an overview of the different techniques to assess REE with special regard to the use of IC in critically ill patients on extracorporeal membrane oxygenation (ECMO), as well as to the sensors used in commercially available indirect calorimeters. The theoretical and technical aspects of IC in spontaneously breathing subjects and critically ill patients on mechanical ventilation and/or ECMO are covered and a critical review and comparison of the different techniques and sensors is provided. This review also aims to accurately present the physical quantities and mathematical concepts regarding IC to reduce errors and promote consistency in further research. By studying IC on ECMO from an engineering point of view rather than a medical point of view, new problem definitions come into play to further advance these techniques.