With the rapid advancement of 5G networks, billions of smart Internet of Things (IoT) devices along with an enormous amount of data are generated at the network edge. While still at an early age, it is expected that the evolving 6G network will adopt advanced artificial intelligence (AI) technologies to collect, transmit, and learn this valuable data for innovative applications and intelligent services. However, traditional machine learning (ML) approaches require centralizing the training data in the data center or cloud, raising serious user-privacy concerns. Federated learning, as an emerging distributed AI paradigm with privacy-preserving nature, is anticipated to be a key enabler for achieving ubiquitous AI in 6G networks. However, there are several system and statistical heterogeneity challenges for effective and efficient FL implementation in 6G networks. In this article, we investigate the optimization approaches that can effectively address the challenging heterogeneity issues from three aspects: incentive mechanism design, network resource management, and personalized model optimization. We also present some open problems and promising directions for future research.