The de Sitter (dS) group and its representations : an introduction to elementary systems and modeling the dark energy universe Introduction Preliminary: 1 + 1 dimensional dS (dS2) Geometry and Relativity 1+3 dimensional dS (dS4) Geometry and Relativity (Classical and Quantum Mechanics) 1+3 dimensional dS (dS4) Geometry and Relativity (QFT) Notion of Mass in (A)dS4 Relativity. This book reviews the construction of elementary systems living in de Sitter (dS) spacetime in both the classical and quantum senses. Field theories on dS spacetime are among the most studied mathematical models of the Universe whether for its earlier period (inflationary phase) or for its current phase of expansion acceleration (dark energy or cosmological constant). Classical elementary systems are Hamiltonian phase spaces which are associated with co adjoint orbits of the relativity group. On the other hand quantum elementary systems are associated with (projective) unitary irreducible representations of the (possibly extended) relativity group (or one of its covering). This study emphasizes the conceptual issues arising in the formulation of such systems and discusses known results in a mathematically rigorous way. Particular attention is paid to: “smooth” transition from classical to quantum theory; physical content under vanishing curvature from the point of view of a local (“tangent”) Minkowskian observer; and thermal interpretation (on the quantum level) in the sense of the Gibbons Hawking temperature. Such a mathematical construction is of paramount importance to the understanding of the early Universe (due to the critical role that the dS metric plays in the inflationary cosmological scenarii) as well as to the construction of possible models for late time cosmology (since a small positive cosmological constant or dark energy seems to be required by recent data). In this sense this book uniquely blends mathematical physics (spacetime symmetry on classical and quantum levels) and theoretical physics (quantization quantum field theory and cosmology). Moreover the level of exposition varies in different parts of the book so that both experts and beginners alike can utilize the book. In addition this book: Presents consistent formulations of dS elementary systems on three levels; classical mechanics quantum mechanics and the quantum field theory Blends mathematical physics (spacetime symmetry on classical and quantum levels) and theoretical physics (quantization quantum field theory and cosmology) to uniquely introduce the mathematical structure of dS relativity Discusses conceptual considerations worries that arise in the formulation of elementary systems.