Is There a Temperature : Conceptual Challenges at High Energy Acceleration and Complexity Temperature and heat entropy and order or disorder are key classical concepts of physics. These are challenged by searching matter under extreme conditions such as high (relativistic) energy strong acceleration or gravitation or unusual complexity due to long range correlations. In our quest for quark matter all these conditions might occur simultaneously. This book strongly motivated by the authors everyday research experiences in the field of high energy heavy ion collisions aims to bundle these challenges to modern physics. The main topic is at the heart of thermodynamics the very concept of temperature its use and extensions. New developments on this issue are both applications and foundations of non extensive statistics as well as concepts borrowed from gravity and string theory to describe the surprisingly statistical behavior of elementary matter at the highest accelerator energies of the world. The reader will benefit from bringing these new developments in one book together by having the view of classical and modern concepts at the heart of physics across the problems related to high energy high acceleration and high complexity. After reviewing the classical approaches the author discusses the dual gravity and non extensive statistical aspects of heavy ion collisions describing these experimental findings with the use of the concept of temperature.