Here, we explore how nanoscale machines and engines work. In biology, molecular machines perform energy conversion, signal transduction, regulation, transcription, ion pumping, molecular transport, and more. A molecular machine can be as simple as a single protein or nucleic acid molecule. How can a single protein or DNA molecule transduce one type of energetic action into another, in repeated cycles? A cyclic transducer is another name for an engine. The key to transduction is coupled binding events. For example, a molecular motor, such as myosin, can create motion because the motor’s binding to a molecular ‘track protein,’ such as actin, is coupled to the motor’s binding to ATP. We explore the principles of biomolecular machines using binding polynomials. We begin with oxygen binding to globin proteins, probably the best understood process of coupled binding and cooperativity in biology. Hemoglobin’s affinity to bind one oxygen molecule depends on how many other oxygens are already bound. Because of this cooperativity, hemoglobin can pick up oxygen in the lungs and drop it off in the tissues. And, oxygen transport can be regulated by pH because hemoglobin’s binding to oxygen is coupled to hemoglobin’s binding to protons.