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The so-called Complexity Sciences are a topic of fast growing interest inside the scientific community. Actually, researchers did not come to a definition of complexity, since it manifests itself in so many different ways [1]. This field itself is not a single discipline, but rather a heterogeneous amalgam of different techniques of mathematics and science. In fact, under the label of Complexity Sciences we comprehend a large variety of approaches: nonlinear dynamics, deterministic chaos theory, nonequilibrium thermodynamics, fractal geometry, intermediate asymptotics, complete and incomplete similarity, renormalization group theory, catastrophe theory, self-organized criticality, neural networks, cellular automata, fuzzy logic, etc. Aim of this paper is at providing insight into the role of complexity in the field of Materials Science and Fracture Mechanics [2-3]. The presented examples will be concerned with the snap-back instabilities in the structural behaviour of composite structures (Carpinteri [4-6]), the occurrence of fractal patterns and self-similarity in material damage and deformation of heterogeneous materials, and the apparent scaling on the nominal mechanical properties of disordered materials (Carpinteri [7,8]). Further examples will deal with criticality in the acoustic emissions of damaged structures and with scaling in the time-to-failure (Carpinteri et al. [9]). Eventually, results on the transition towards chaos in the dynamics of cracked beams will be reported (Carpinteri and Pugno [10,11]).