Satellite image time series (SITS) is a sequence of satellite images that record a given area at several consecutive times. The aim of such sequences is to use not only spatial information but also the temporal dimension of the data, which is used for multiple real-world applications, such as classification, segmentation, anomaly detection, and prediction. Several traditional machine learning algorithms have been developed and successfully applied to time series for predictions. However, these methods have limitations in some situations, thus deep learning (DL) techniques have been introduced to achieve the best performance. Reviews of machine learning and DL methods for time series prediction problems have been conducted in previous studies. However, to the best of our knowledge, none of these surveys have addressed the specific case of works using DL techniques and satellite images as datasets for predictions. Therefore, this paper concentrates on the DL applications for SITS prediction, giving an overview of the main elements used to design and evaluate the predictive models, namely the architectures, data, optimization functions, and evaluation metrics. The reviewed DL-based models are divided into three categories, namely recurrent neural network-based models, hybrid models, and feed-forward-based models (convolutional neural networks and multi-layer perceptron). The main characteristics of satellite images and the major existing applications in the field of SITS prediction are also presented in this article. These applications include weather forecasting, precipitation nowcasting, spatio-temporal analysis, and missing data reconstruction. Finally, current limitations and proposed workable solutions related to the use of DL for SITS prediction are also highlighted.