Federated learning (FL) is an exciting machine learning approach where multiple devices collaboratively train a model without sharing their raw data. The FL system is vulnerable to the action of Byzantine clients sending arbitrary model updates, and the trained model may exhibit prediction bias towards specific groups. However, FL mechanisms tackling robustness and bias mitigation have contradicting objectives, motivating the question of building a FL system that comprehensively combines both objectives. In this paper, we first survey state-of-the-art approaches to robustness to Byzantine behavior and bias mitigation and analyze their respective objectives. Then, we conduct an empirical evaluation to illustrate the interplay between state-of-the-art FL robustness mechanisms and FL bias mitigation mechanisms. Specifically, we show that classical robust FL methods may inadvertently filter out benign FL clients that have statistically rare data, particularly for minority groups. Finally, we derive research directions for building more robust and bias-free FL systems.