Responsible AI becomes critical where robustness and fairness must be satisfied together. Traditionally, the two topics have been studied by different communities for different applications. Robust training is designed for noisy or poisoned data where image data is typically considered. In comparison, fair training primarily deals with biased data where structured data is typically considered. Nevertheless, robust training and fair training are fundamentally similar in considering that both of them aim at fixing the inherent flaws of real-world data. In this tutorial, we first cover state-of-the-art robust training techniques where most of the research is on combating various label noises. In particular, we cover label noise modeling, robust training approaches, and real-world noisy data sets. Then, proceeding to the related fairness literature, we discuss pre-processing, in-processing, and post-processing unfairness mitigation techniques, depending on whether the mitigation occurs before, during, or after the model training. Finally, we cover the recent trend emerged to combine robust and fair training in two flavors - the former is to make the fair training more robust (i.e., robust fair training), and the latter is to consider robustness and fairness as two equals to incorporate them into a holistic framework. This tutorial is indeed timely and novel because the convergence of the two topics is increasingly common, but yet to be addressed in tutorials. The tutors have extensive experience publishing papers in top-tier machine learning and data mining venues and developing machine learning platforms.