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Abstract A fundamental problem in functional data analysis is to classify a functional observation based on training data. The application of functional data classification has gained immense popularity and utility across a wide array of disciplines, encompassing biology, engineering, environmental science, medical science, neurology, social science, and beyond. The phenomenal growth of the application of functional data classification indicates the urgent need for a systematic approach to develop efficient classification methods and scalable algorithmic implementations. Therefore, we here conduct a comprehensive review of classification methods for functional data. The review aims to bridge the gap between the functional data analysis community and the machine learning community, and to intrigue new principles for functional data classification. This article is categorized under:Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and ClassificationStatistical Models > Classification ModelsData: Types and Structure > Time Series, Stochastic Processes, and Functional Data