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This paper reviews recent research works in infant cry signal analysis and classification tasks. A broad range of literatures are reviewed mainly from the aspects of data acquisition, cross domain signal processing techniques, and machine learning classification methods. We introduce pre-processing approaches and describe a diversity of features such as MFCC, spectrogram, and fundamental frequency, etc. Both acoustic features and prosodic features extracted from different domains can discriminate frame-based signals from one another and can be used to train machine learning classifiers. Together with traditional machine learning classifiers such as KNN, SVM, and GMM, newly developed neural network architectures such as CNN and RNN are applied in infant cry research. We present some significant experimental results on pathological cry identification, cry reason classification, and cry sound detection with some typical databases. This survey systematically studies the previous research in all relevant areas of infant cry and provides an insight on the current cutting-edge works in infant cry signal analysis and classification. We also propose future research directions in data processing, feature extraction, and neural network classification fields to better understand, interpret, and process infant cry signals. [ABSTRACT FROM AUTHOR]