Background Little is known about the incidence of severe critical events in children undergoing general anaesthesia in Europe. We aimed to identify the incidence, nature, and outcome of severe critical events in children undergoing anaesthesia, and the associated potential risk factors. Methods The APRICOT study was a prospective observational multicentre cohort study of children from birth to 15 years of age undergoing elective or urgent anaesthesia for diagnostic or surgical procedures. Children were eligible for inclusion during a 2-week period determined prospectively by each centre. There were 261 participating centres across 33 European countries. The primary endpoint was the occurence of perioperative severe critical events requiring immediate intervention. A severe critical event was defined as the occurrence of respiratory, cardiac, allergic, or neurological complications requiring immediate intervention and that led (or could have led) to major disability or death. This study is registered with ClinicalTrials.gov, number NCT01878760. Findings Between April 1, 2014, and Jan 31, 2015, 31 127 anaesthetic procedures in 30 874 children with a mean age of 6·35 years (SD 4·50) were included. The incidence of perioperative severe critical events was 5·2% (95% CI 5·0–5·5) with an incidence of respiratory critical events of 3·1% (2·9–3·3). Cardiovascular instability occurred in 1·9% (1·7–2·1), with an immediate poor outcome in 5·4% (3·7–7·5) of these cases. The all-cause 30-day in-hospital mortality rate was 10 in 10 000. This was independent of type of anaesthesia. Age (relative risk 0·88, 95% CI 0·86–0·90; p<0·0001), medical history, and physical condition (1·60, 1·40–1·82; p<0·0001) were the major risk factors for a serious critical event. Multivariate analysis revealed evidence for the beneficial effect of years of experience of the most senior anaesthesia team member (0·99, 0·981–0·997; p<0·0048 for respiratory critical events, and 0·98, 0·97–0·99; p=0·0039 for cardiovascular critical events), rather than the type of health institution or providers. Interpretation This study highlights a relatively high rate of severe critical events during the anaesthesia management of children for surgical or diagnostic procedures in Europe, and a large variability in the practice of paediatric anaesthesia. These findings are substantial enough to warrant attention from national, regional, and specialist societies to target education of anaesthesiologists and their teams and implement strategies for quality improvement in paediatric anaesthesia. Funding European Society of Anaesthesiology.