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In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach. [ABSTRACT FROM AUTHOR]

Jacobi groupoids are introduced as a generalization of Poisson and contact groupoids and it is proved that generalized Lie bialgebroids are the infinitesimal invariants of Jacobi groupoids. Several examples are discussed. [Copyright &y& Elsevier]

Lichnerowicz–Jacobi cohomology of Jacobi manifolds is reviewed. The use of the associated Lie algebroid allows to prove that the Lichnerowicz–Jacobi cohomology is invariant under conformal changes of the Jacobi structure. We also compute the Lichnerowicz–Jacobi cohomology for a large variety of examples. [Copyright &y& Elsevier]