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In this book the author explores the hypothesis that quantum effects of spacetime may manifest through the non-commutativity of the spacetime itself at short distances. The focus is on the two most studied examples of quantum-non-commutative spacetimes: canonical spacetime and k-Minkowski spacetime. In the first part of the book the author analyzes issues related to the propagation of classical waves on these quantum spacetimes. In the second part of the book the author discusses quantum field theory on canonical spacetime focusing on issues, related to the so-called IR/UV mixing, which are relevant for the phenomenology.