Recently the concept of transformative AI (TAI) has begun to receive attention in the AI policy space. TAI is often framed as an alternative formulation to notions of strong AI (e.g. artificial general intelligence or superintelligence) and reflects increasing consensus that advanced AI which does not fit these definitions may nonetheless have extreme and long-lasting impacts on society. However, the term TAI is poorly defined and often used ambiguously. Some use the notion of TAI to describe levels of societal transformation associated with previous 'general purpose technologies' (GPTs) such as electricity or the internal combustion engine. Others use the term to refer to more drastic levels of transformation comparable to the agricultural or industrial revolutions. The notion has also been used much more loosely, with some implying that current AI systems are already having a transformative impact on society. This paper unpacks and analyses the notion of TAI, proposing a distinction between TAI and radically transformative AI (RTAI), roughly corresponding to societal change on the level of the agricultural or industrial revolutions. We describe some relevant dimensions associated with each and discuss what kinds of advances in capabilities they might require. We further consider the relationship between TAI and RTAI and whether we should necessarily expect a period of TAI to precede the emergence of RTAI. This analysis is important as it can help guide discussions among AI policy researchers about how to allocate resources towards mitigating the most extreme impacts of AI and it can bring attention to negative TAI scenarios that are currently neglected.