Earth’s magnetic field is generated by fluid motions in the outer core. This geodynamo has operated for over 3.4 billion years. However, the mechanism that has sustained the geodynamo for over 75% of Earth’s history remains debated. In this Review, we assess the mechanisms proposed to drive the geodynamo (precession, tides and convection) and their ability to match geomagnetic and palaeomagnetic observations. Flows driven by precession are too weak to drive the geodynamo. Flows driven by tides could have been strong enough in the early Earth, before 1.5 billion years ago, when tidal deformation and Earth’s spin rate were larger than they are today. Evidence that the thermal conductivity of Earth’s core could be as high as 250 W m−1 K−1 calls the ability of convection to maintain the dynamo for over 3.4 billion years into question. Yet, convection could supply enough power to sustain a long-lived geodynamo if the thermal conductivity is lower than 100 W m−1 K−1. Exsolution of light elements from the core increases this upper conductivity limit by 15% to 200%, based on the exsolution rates reported so far. Convection, possibly aided by the exsolution of light elements, remains the mechanism most likely to have sustained the geodynamo. The light-element exsolution rate, which remains poorly constrained, should be further investigated.