This field of physics was originally identified as Solid state Physics, then P.W. Anderson coined the term Condensed Matter Physics and more recently it has merged with Statistical Physics to lead to the Physics of Complex Systems. The study of complex systems refers to the emergency of collective properties in systems with a large number of parts in interaction among them. These elements can be atoms or macromolecules in a physical or biological context, but also people, machines or companies in a socio-economic context. The science of complexity tries to discover the nature of the emerging behavior of complex systems, often invisible to the traditional approach, by focusing on the structure of the interconnections and the general architecture of systems, rather than on the individual components. It is a change of perspective in the forma mentis of scientists rather than a new scientific discipline. Traditional science is based on a reductionistic reasoning for which, if one knows the basic elements of a system, it is possible to predict its behavior and properties. It is easy to realize, however, that for a cell or for the socio-economic dynamics one faces a new situation in which the knowledge of the individual parts is not sufficient to describe the global behavior of the structure. We can represent this situation as the study of the architecture of matter and nature. It depends in some way from the individual elements (bricks) but then it shows fundamental laws and properties which cannot be derived from these elements. Starting from the simplest physical systems, like critical phenomena in which order and disorder compete, these emergent behaviors can be identified in many other systems, from ecology to the immunitary system, to the social behavior and economics. The science of complexity has the objective of understand the properties of these systems. Which rules govern their behavior? How they adapt to changing conditions? How they learn efficiently and ...