Preservation of cognitive function is one of the major concerns in contemporary ageing societies. At the same time, overweight and obesity, which have been identified as risk factors for poor health development, have been increasing in many countries all over the world. This study examines the relationship between bodyweight change and cognitive decline in old age and it aims to determine whether and how changes in body mass index (BMI) affect the development of cognitive functioning in old age. Using longitudinal data from the Survey of Health, Ageing and Retirement in Europe (SHARE), covering four waves between 2006 and 2016 with 58,389 participants from 15 countries aged 50+, we estimated asymmetric fixed effects models by gender, adding possible confounding variables such as age, grip strength, health conditions, and physical activity. Additionally, we investigated possible heterogeneity in the BMI-cognition relation. We found a positive association between BMI change and change in cognitive performance, which was dominantly driven by BMI decrease. Weight loss was typically negatively related to cognition, particularly at low levels of BMI and mainly due to health conditions affecting both bodyweight and cognitive performance. Weight gain was, on average, not significantly related to cognitive performance; only respondents with preceding weight loss profited from small increases in BMI. Our analyses provide no support for an "obesity paradox" in cognition, according to which higher weight preserves cognition in old age. The association between weight change and cognitive performance in older age is based on weight changes being related to illness and recovery.