This book is devoted to a domain of highest industrial and scienti'c interest, the complexity. The complexity understanding and management will be a main source of e'ciency and prosperity for the next decades. Complex systems areassembliesof multiple subsystemsand arecharact- ized by emergent behavior that results by nonlinear interactions among the subsystems at multiple levels of organization. Evolvability that is the ability to evolve is the method to confront and surpass the successive boundaries of complexity. Evolvability is not biological but should be considered here in the sense that the corresponding systems have, at di'erent levels, charact- istics that are naturally associated to the living systems. The signi'cance of the complexity and the phenomena of emergence are highlighted in the ?rst chapterofthe book.Theimplicationofconcepts aslevelofreality, circularity and closure for evolvable systems is evaluated. The second chapter of the book exposes the methodology to analyze and manage complex systems.

The polystochastic models, PSMs, are the cons- ered mathematical tools. PSMs characterize systems emerging when several stochastic processes occurring at di'erent conditioning levels, are capable to interact with each other, resulting in qualitatively new processes and s- tems. Innovative are the higher categories approach and the introduction of apartialdi'erentialmodelfor multiple levelsmodeling.This imposes making use of appropriate notions of time, space, probabilities and entropy. Categorytheoryistheformalismcapabletooutlinethegeneralframework, shared by the functional organization of biological organisms, of cognitive systems, by the operational structure of evolvable technologies and devices and after all by the scienti'c and engineering methods.