We present a new approach for modelling grasping using an integrated space of grasps and shapes. In particular, we introduce an infinite dimensional space, the Grasp Moduli Space, which represents shapes and grasps in a continuous manner. We define a metric on this space allowing us to formalize ‘nearby’ grasp/shape configurations and we discuss continuous deformations of such configurations. We work in particular with surfaces with cylindrical coordinates and analyse the stability of a popular $L_1$ grasp quality measure $Q_l$ under continuous deformations of shapes and grasps. We experimentally determine bounds on the maximal change of $Q_l$ in a small neighbourhood around stable grasps with grasp quality above a threshold. In the case of surfaces of revolution, we determine stable grasps which correspond to grasps used by humans and develop an efficient algorithm for generating those grasps in the case of three contact points. We show that sufficiently stable grasps stay stable under small deformations. For larger deformations, we develop a gradient-based method that can transfer stable grasps between different surfaces. Additionally, we show in experiments that our gradient method can be used to find stable grasps on arbitrary surfaces with cylindrical coordinates by deforming such surfaces towards a corresponding ‘canonical’ surface of revolution.