Quantum Information Processing (QIP) performs wonders in a world that obeys the laws of quantum mechanics, whereas Machine Learning (ML) is generally assumed to be done in a classical world. We initiate an investigation of the encounter of ML with QIP by defining and studying novel learning tasks that correspond to Machine Learning in a world in which the information is fundamentally quantum mechanical. We shall see that this paradigm shift has a profound impact on the learning process and that our classical intuition is often challenged.