Abstract: We attempt to explain the manifold settings ("curve" spaces) on which we can study the theory of elliptic operators just like in the local setting (ℝn). In particular, we are interested in the analysis of Dirac operators D2 and the Laplacian D2 (they are first and second order elliptic operators respectively) associated to a given smooth oriented closed manifold. We note that though locally on a smooth oriented closed manifold, one can always define the Dirac operators, a global definition of such operators might not exist. The precise ingredient needed for the global picture of Dirac operators is the existence of Clifford bundles over a manifold. The study of Clifford bundles over a manifold—specifically spinor bundles over a spin manifold—and their associated (globally defined) Dirac operators is of importance as illustrated by, say, the beautiful index theorems of Atiyah–Singer that give a meaningful relationship between topology, geometry and analysis.