Unlike certain personal or national tragedies which may extend indefinitely, patterns observed in nature are of finite extent. Yet, as a rule, the solitary patterns predicted by almost all existing mathematical models extend indefinitely with their tails being a by product of their analytical nature. Rather then viewing such tails as a manifestation of the inherent limitation of math to model physics in detail, we adopt the opposite view: the persistence of tails in a large variety of solitary patterns points to a missing mechanism capable to constrain the pattern. Clearly, to induce a compact pattern one has to escape the curse of analyticity. Differently stated, one has to supplement the existing models with a mechanism(s) which may beget a local singularity. When this is done the resulting local loss of solution's uniqueness enables to connect a smooth part of the solution with the trivial ground state and thus to form an entity with a compact support: the compacton. We shall describe a variety of singularity inducing mechanisms that beget compact solutions of dispersive or dissipative uni and multi-dimensional phenomena. Compactified variants of the K-dV, Klein-Gordon and Schroedinger equations will be surveyed. In Part two of the lecture we shall discuss the intriguing nature of these (weakly strong or strongly weak) solutions, the underlying singularities and their relation with a discrete antecedent where a sharp fronts are replaced with tails decaying at a doubly exponential rate.