Abstract Generative artificial intelligence (GenAI) models have attracted tremendous interest since the advent of ChatGPT, raising numerous opportunities and challenges. However, their generative power has not yet been studied, leaving open the question of what is truly generated by these tools. This paper addresses this question and precisely characterizes the generativity behind GenAI models. Owing to the latest advancements in engineering design, we first propose a framework for uncovering the various types of generativity. Then, we consider the main families of GenAI models and systematically analyze them to characterize their generativity within this framework. By doing so, we highlight the existence of two distinct generative levels in GenAI: one leading to the generation of new artifacts and the other leading to the generation of GenAI models themselves. We are also able to characterize the generativity of both of these levels, thus specifically confirming the generative power of GenAI and opening research avenues toward human-GenAI collaboration.