Quantum versus Chaos : Questions Emerging from Mesoscopic Cosmos Quantum and chaos key concepts in contemporary science are incompatible by nature. This volume presents an investigation into quantum transport in mesoscopic or nanoscale systems which are classically chaotic and shows the success and failure of quantal semiclassical and random matrix theories in dealing with questions emerging from the mesoscopic cosmos. These traditional theories are critically analysed and this leads to a new direction. To reconcile quantum with chaos and to restore genuine temporal chaos in quantum systems a time discrete variant of quantum dynamics is proposed. Audience:This book will be of interest to graduate students and researchers in physics chemistry and mathematics whose work involves fundamental questions of quantum mechanics in chaotic systems. Genesis of Chaos and Breakdown of Quantization of Adiabatic Invariants Semiclassical Quantization of Chaos: Trace Formula Pseudo Chaos Without Classical Counterpart in One Dimensional Quantum Transport Chaos and Quantum Transport in Open Magnetic Billiards: from Stadium to Sinai Billiards Chaotic Scattering on Hyperbolic Billiards: Success of Semiclassical Theory Nonadiabaticity Induced Quantum Chaos Level Dynamics and Statistical Mechanics Towards Time Discrete Quantum Mechanics Conclusions and Prospects.