Rydberg atoms with exaggerated properties (enormous sizes, very long life times) can be prepared and manipulated by laser and radiofrequency excitation with an exquisite precision. They interact very strongly with microwave photons and between each other, at distances which are huge at the atomic scale. These features make them ideal tools to explore fundamental quantum phenomena, to build quantum gates and to realize quantum simulators of condensed matter systems. In the first part of the talk I will recall the early history of Rydberg atom physics and the Cavity Quantum Electrodynamics experiments which have made possible the entanglement of atoms, the non-destructive manipulation and detection of single photons and the study of Schrödinger cat states of light. In the second part, I will describe recent studies in which arrays of Rydberg atoms are interacting with each other in a controlled way, opening promising perspectives in quantum information science. Extension of these studies to artificial atoms of even bigger size made of superconducting circuits will also be mentioned.