In recent years there have been an increasing number of results where quantum physics has been combined with machine learning for different reasons. On the one hand, quantum computers promise to significantly speed up some of the computational techniques used in machine learning and, on the other hand, “classical” machine learning methods can help us with the verification and classification of complex quantum systems. Moreover, the rich mathematical structure of quantum mechanics can help define new models and learning paradigms. In this talk, we will introduce quantum machine learning in all of these flavors, and then discuss how to bound the accuracy and generalization errors via entropic quantities. These bounds establish a link between the compression of information into quantum states and the ability to learn, and allow us to understand how difficult it is, namely how many samples are needed in the worst case scenario, to learn a quantum classification problem from examples. Different applications will be considered, such as the classification of complex phases of matter, entanglement classification, and the optimization of quantum embeddings of classical data.