This paper demonstrates that the element-free Galerkin (EFG) method can be successfully used in shape design sensitivity analysis and shape optimization for problems in 2D elasticity. The continuum-based variational equations for displacement sensitivities are derived and are subsequently discretized. This approach allows one to avoid dierentiating the EFG shape functions. The present formulation, that employs a penalty method for imposing the essential boundary conditions, can be easily extended to 3D and/or non-linear problems. Numerical examples are presented to show the capabilities of the current approach for calculating sensitivities. The ยฏexibility of the EFG method, that eliminates the element connectivity requirement of the ยฎnite element method (FEM), permits solving shape optimization problems without re-meshing. The problem of shape optimization of a ยฎllet is used to demonstrate this fact. Smoother stresses and better accuracy for points close to the boundary allow for a better EFG solution compared to published results using the FEM, the boundary element method (BEM) or the boundary contour method (BCM). Furthermore, for the EFG approach, grid-optimization appears unnecessary.