Foundation models have emerged as a dominant paradigm in machine learning, enabling broad generalization and efficient adaptation across diverse tasks and domains. While this paradigm has achieved remarkable success in language and vision data, its extension to structured data remains far less understood. Foundation models for structured data are an emerging yet highly impactful research area with a rapidly growing body of literature. In this survey, we provide a systematic analysis of foundation models for structured data, focusing on tabular, time series, and graph data, covering over 150 representative methods. We analyze the intrinsic properties and inductive biases of structured data, clarify the core concepts of foundation models, and conduct an in-depth analysis of the key challenges that hinder the development of foundation models for structured data. Building on these insights, we organize existing approaches into a coherent taxonomy based on tokenization, architectures, pre-training objectives, and adaptation strategies. Finally, we discusse merging research directions and open problems, aiming to provide guidance toward more principled and scalable foundation models for structured data.