Why Symmetry? -- J-Pairing Interractions of Fermions in a Single-J Shell -- Supersymmetry in Nuclei -- The Relativistic Many Body Problem in Quantum Mechanics -- Gauge Invariance and the E1 Sum Rule in Nuclei -- The Heisenberg Group in Classical and Quantum Information Transmission -- Quantum Field Theory of Particle Mixing and Oscillations -- Two-Photon Interactions in Cavity QED -- Low-Dimensional Spin Systems: Hidden Symmetries, Conformal Field Theories and Numerical Checks -- Quantum Tomography, Wave Packets and Solitons -- Borel Quantization and Nonlinear Quantum Mechanics -- Seeing Science through Symmetry -- Space-Time Symmetries on Clifford Algebra C4 Based Representation Spaces of su(4) ? so(6) -- Fermionic O(8) and Bosonic U(36) Symmetry Schemes for Heavy N=Z Nuclei -- Interaction and Fusion of Elementary Systems -- Propagation in Crossed Electric and Magnetic Fields: The Quantum Source Approach -- On Group Theoretical Aspects, Hypergeometric Transformations and Symmetries of Angular Momentum Coefficients -- Tensor and Spin Representations of SO(4) and Discrete Quantum Gravity -- The Geometry of Density States, Positive Maps and Tomograms -- Objective Existence and Relativity Groups -- Survival of Quasi-Spin Structure in Isomers of N ? 82 Nuclei -- Irreversible Wave Packet Dynamics in a Magnetic Field -- From Quantum Groups to Genetic Mutations -- Noncompact Quantum Algebra Uq(2,1): Positive Discrete Series of Irreducible Representations -- Combinatorial Physics, Normal Order and Model Feynman Graphs -- Bohr’s Symmetry and the Quantum Numbers for the Triaxially Superdeformed Bands in Odd Mass Nuclei -- Understanding Brain and Consciousness? -- Beta-Decay in Odd-A Cs to Xe in the Interacting Boson-Fermion Model -- Triaxiality and Chirality in Nuclei Around Mass 130.This book is a collection of reviews and essays about the recent developments in the area of Symmetries and applications of Group Theory. Contributions have been written mostly at the graduate level but some are accessible to advanced undergraduates. The book is of interest to a wide audience and covers a broad range of topics with a strong degree of thematical unity. The book is part of a Series of books on Symmetries in Science and may be compared to the published Proceedings of the Colloquia on Group Theoretical Methods in Physics. Here, however, prevails a distinguished character for presenting extended reviews on present applications to Science, not restricted to Theoretical Physics.