This dissertation describes the rigidity theory of bar joint frameworks, especially infinite ones. The first chapter revises some of the well established results for finite frameworks. We then look at how this can be extended to the infinite case, specifically from the analysis point of view. In particular, we look at vanishing flexibility that is observed in some specific examples. Then we look at a proof of the sufficient condition for the existence of a flex in an infinite framework as described in Owen and Power [6]. In the fourth chapter we establish that the rigidity operator arising from the infinite matrix is bounded. 'Ve then observe its structure for specific examples. As decribed in [8], we describe the representation of the rigidity operator as a matrix valued function on the torus. Finally we look at the decomposition of the space of infinitesimal flexes for crystal frameworks in terms of a product basis.