Recently, domain adaptation (DA) has been widely used in the remaining useful life (RUL) prediction of rotating machinery to effectively mitigate domain shift. Traditional DA methods for RUL prediction mainly focus on single-source domain adaptation (SDA) algorithms. However, labeled data can often be collected from multiple sources in practical scenarios. Directly applying SDA algorithms may degrade the model performance. Therefore, this paper proposes a novel multi-source adversarial distillation domain adaptation (MADDA) network for RUL regression problems. Specifically, a source feature extractor and regressor are pre-trained for each labeled source domain to capture source-specific representation. Then, a target encoder is learned to align target and source features via adversarial training to alleviate domain shift. Furthermore, a source distillation weighting mechanism is devised to utilize source samples that are more similar to target domains for fine-tuning the source regressor, thereby enhancing its performance on target tasks. Meanwhile, a source aggregation strategy is proposed to assign domain weights to the prediction results of various source regressors depending on the disparities between the source and the target domain, aiming to achieve the optimal combination of the final prediction. Case studies on two bearing datasets demonstrate the effectiveness and superiority of the proposed method.