Artificial intelligence methods and application have recently shown great contribution in modeling and prediction of the hydrological processes, climate change, and earth systems. Among them, deep learning and machine learning methods mainly have reported being essential for achieving higher accuracy, robustness, efficiency, computation cost, and overall model performance. This paper presents the state of the art of machine learning and deep learning methods and applications in this realm and the current state, and future trends are discussed. The survey of the advances in machine learning and deep learning are presented through a novel classification of methods. The paper concludes that deep learning is still in the first stages of development, and the research is still progressing. On the other hand, machine learning methods are already established in the fields, and novel methods with higher performance are emerging through ensemble techniques and hybridization.