In the last years, many European countries have experienced the effects of climate change, in the form of a scarcity of drinking water resources, prolonged periods of drought, and extremely heavy rainfall, with unprecedented dramatic environmental, economic, and social costs. Therefore, understanding, modeling, and predicting the movement and distribution of water on Earth, and effectively managing water resources are problems of paramount importance. In this article, we discuss the fundamental role that sensing technologies, data processing algorithms, and inference based on machine learning techniques can have in modern hydrology and the many challenges that still need to be addressed to improve the accuracy and reduce the complexity of current hydrology models. More specifically, we overview the main solutions proposed in the literature to monitor, analyze and predict hydrological processes, and present a selection of results obtained from empirical data sets to ground the main concepts and substantiate the dissertation. Finally, we conclude our article by discussing open problems and possible avenues for future research.