Generative artificial intelligence (GenAI) has quickly become prolific in our daily lives, including the higher education sector. Although an AI-fuelled world is unpredictable, there is an urgent need to understand how university students use GenAI to support their learning and the factors influencing GenAI adoption. In this study, underpinned by self-regulated learning (SRL) theory and the technology acceptance model, we examined how university students' adoption of GenAI is influenced by their SRL skills. Given the importance of SRL skills on students' selection of study strategies, we hypothesised that SRL constructs would have a strong association with their adoption of GenAI. To explore this, we conducted an international survey-based study of 435 students from two universities in Australia and Canada to capture students' use of GenAI within the first year of its wider adoption. Our findings reveal that SRL constructs of self-efficacy and social support predict the perceived ease of use of GenAI. Intrinsic motivation and effort regulation also predicted the perceived usefulness of GenAI, with a stronger association for students using GenAI for university learning than those who used it for non-academic purposes such as work or personal use. We discuss implications of our findings for educators.