Streaming data continuously generated by real-world systems and have become a central research topic in data mining and machine learning due to their broad applications in transportation, environmental monitoring, and social networks. Unlike static offline data, streaming data arrive continuously and often exhibit non-stationary characteristics, including temporal distribution drift, spatial topology evolution, and dynamic spatio-temporal coupling. These properties pose significant challenges to robust modeling, continual adaptation, and effective evaluation. This survey provides a comprehensive review of streaming data modeling, spanning the evolution from conventional deep learning to foundation models and large language models. We first introduce the basic forms of streaming data, including time series streams, dynamic graph streams, and spatio-temporal streams. We then organize existing studies along three core dimensions: data adaptation, model design, and task taxonomy. This perspective highlights how continuous observations drive data-level adaptation, how model architectures are designed to capture the dynamic characteristics of streaming data, and how tasks in streaming data scenarios are formally defined, categorized, and evaluated. In addition, we summarize representative applications, public datasets, and benchmark resources. Finally, we outline current challenges in streaming data mining and modeling and discuss future research directions.