This paper presents a new mesh generation method for a convex polygonal domain. We first decompose the convex polygon into simple sub regions in the shape of triangles. These simple regions are then triangulated to generate a mesh of 6-node triangular elements. We propose then an automatic 6-node trianglar to 3-node triangular conversion scheme. Each isolated 6-node triangle is split into four triangles according to the usual scheme,that is,by adding three vertices in the middle of the edges and joining them by straight lines. To preserve the mesh conformity a similar procedure is also applied to every triangle of the domain to fully discretize the given convex polygonal domain into all triangles, thus propagating uniform refinement. This simple method generates a high quality mesh whose elements confirm well to the requested shape by refining the problem domain. Examples are presented to illustrate the simplicity and efficiency of the new mesh generation method for standard and arbitrary shaped polygonal domains and cracked polygonal domains. We have appended MATLAB programs which incorporate the mesh generation scheme developed in this paper. These programs provide valuable output on the nodal coordinates ,element connectivity and graphic display of the all triangular mesh for application to finite element analysis.