Despite the importance of grasp synthesis for autonomous robotic operations, its formulation for continuum robots (CRs) has yet to be investigated. This paper presents a grasp taxonomy and a synthesis approach for CRs. The proposed synthesis relies on an analytical model for grasping using CRs. The method is extended for the application of cooperative continuum robots (CCRs). To present a comprehensive formulation of the problem, both constant-curvature and Cosseratrod models are adopted in the proposed grasp synthesis. A set of grasp quality measures are formulated for this study, in which a new quantitative grasp quality measure is introduced to reflect the limited workspace of CRs. Finally, two experimental grasp quality measures are introduced (i.e., path-following error and grasp success rate) to compare and assess the grasps using statistical tests. The effectiveness of the proposed methodology is shown through extensive simulations and experiments using single-segment tendon-driven catheters.