We introduce a new framework of adversarial attacks, named calibration attacks, in which the attacks are generated and organized to trap victim models to be miscalibrated without altering their original accuracy, hence seriously endangering the trustworthiness of the models and any decision-making based on their confidence scores. Specifically, we identify four novel forms of calibration attacks: underconfidence attacks, overconfidence attacks, maximum miscalibration attacks, and random confidence attacks, in both the black-box and white-box setups. We then test these new attacks on typical victim models with comprehensive datasets, demonstrating that even with a relatively low number of queries, the attacks can create significant calibration mistakes. We further provide detailed analyses to understand different aspects of calibration attacks. Building on that, we investigate the effectiveness of widely used adversarial defences and calibration methods against these types of attacks, which then inspires us to devise two novel defences against such calibration attacks.