Trustworthy machine learning (TML) under imperfect data has recently brought much attention in the data-centric fields of machine learning (ML) and artificial intelligence (AI). Specifically, there are mainly three types of imperfect data along with their challenges for ML, including i) label-level imperfection: noisy labels; ii) feature-level imperfection: adversarial examples; iii) distribution-level imperfection: out-of-distribution data. Therefore, in this paper, we systematically share our insights and solutions of TML to handle three types of imperfect data. More importantly, we discuss some new challenges in TML, which also open more opportunities for future studies, such as trustworthy foundation models, trustworthy federated learning, and trustworthy causal learning.