Artificial intelligence (AI) promises to change health care, with some studies showing proof of concept of a providerlevel performance in various medical specialties. However, there are many barriers to implementing AI, including patient acceptance and understanding of AI. Patients' attitudes toward AI are not well understood. We systematically reviewed the literature on patient and general public attitudes toward clinical AI (either hypothetical or realised), including quantitative, qualitative, and mixed methods original research articles. We searched biomedical and computational databases from Jan 1, 2000, to Sept 28, 2020, and screened 2590 articles, 23 of which met our inclusion criteria. Studies were heterogeneous regarding the study population, study design, and the field and type of AI under study. Six (26%) studies assessed currently available or soon-to-be available AI tools, whereas 17 (74%) assessed hypothetical or broadly defined AI. The quality of the methods of these studies was mixed, with a frequent issue of selection bias. Overall, patients and the general public conveyed positive attitudes toward AI but had many reservations and preferred human supervision. We summarise our findings in six themes: AI concept, AI acceptability, AI relationship with humans, AI development and implementation, AI strengths and benefits, and AI weaknesses and risks. We suggest guidance for future studies, with the goal of supporting the safe, equitable, and patient-centred implementation of clinical AI.