Due to the rise of continuous data-generating applications, analyzing data streams has gained increasing attention over the past decades. A core research area in stream data is stream classification, which categorizes or detects data points within an evolving stream of observations. Areas of stream classification are diverse—ranging, e.g., from monitoring sensor data to analyzing a wide range of (social) media applications. Research in stream classification is related to developing methods that adapt to the changing and potentially volatile data stream. It focuses on individual aspects of the stream classification pipeline, e.g., designing suitable algorithm architectures, an efficient train and test procedure, or detecting so-called concept drifts. As a result of the many different research questions and strands, the field is challenging to grasp, especially for beginners. This survey explores, summarizes, and categorizes work within the domain of stream classification and identifies core research threads over the past few years. It is structured based on the stream classification process to facilitate coordination within this complex topic, including common application scenarios and benchmarking data sets. Thus, both newcomers to the field and experts who want to widen their scope can gain (additional) insight into this research area and find starting points and pointers to more in-depth literature on specific issues and research directions in the field.