The present paper deals with the development of a simple meshless method, known as element-free Galerkin method (EFG), and its numerical implementation and application for the solution of 3-D elastic fracture mechanics problems. Meshless methods are rather new computational techniques that do not require the use of any connectivity concept, such as those used in the finite element method (FEM); since only a cloud of nodes is required, the EFG method is particularly suitable for problems involving internal boundaries, geometry changes, and so on. In the present paper the development of the EFG method and its numerical implementation to 3-D linear elasticity is presented with emphasis to the solution of problems with geometric discontinuities such as cracks. The description of the 3-D body is performed by simply employing triangles in space to describe edges (external or internal) and by generating a grid of internal points; if required, a local cloud of nodes concentrated around high stressed zones can be added to the model. The ‘‘visibility criterion’’ is used to detect internal or external boundaries and a Gauss-type weight function, together with a penalty technique, is employed to enforce the boundary conditions. With the developed EFG method some fundamental 3-D fracture mechanics problems are solved in order to verify the computational capability and accuracy of the method.