Intrinsically disordered proteins (IDPs) are widespread in eukaryotic proteomes and challenge the classical structure-function paradigm that equates a folded 3-D structure with protein function. However, IDPs often function by molecular recognition, in which they bind a partner molecule and undergo "induced folding" or "disorder-to-order transition" upon binding, which apparently suggests that in a functional context IDPs become ordered. Whereas this observation would restore the "prestige" of the classical structure-function paradigm, a closer inspection of the complexes of IDPs reveals that they do not always become fully ordered, but preserve functionally significant disorder in the complex with their binding partner(s). This phenomenon, which we termed "fuzziness", is the ultimate extension of structural disorder to the functional native state of proteins. In this introductory chapter, we outline the most important aspects of fuzziness, such as its structural categories, molecular mechanisms of function it mediates and the biological processes, in which it plays a distinguished role. As confirmed by all the other chapters of the book, we will show that new cases of fuzziness pop up at an accelerating pace, underscoring that this phenomenon presents a widespread novel paradigm of protein structure and function.