A new approach for the automatic generation and refinement of finite element meshes over multiply connected planar regions has been developed. This paper represents continuation of authors research activities in that area. An algorithm for producing a triangular mesh in a convex polygon is presented in authors recent work. It is used for the finite element triangulation of a complex polygonal region of the plane decomposed into convex polygons. We decompose the convex polygonal regions into simple sub regions in the shape of triangles. These simple regions are then triangulated to generate a fine mesh of triangular elements. We then propose an automatic triangular to quadrilateral conversion scheme.In this scheme, each isolated triangle is split into three quadrilaterals according to the usual scheme, adding three vertices in the middle of the edges and a vertex a the barycentre of the element. To preserve the mesh conformity, a similar procedure is also applied to every triangle of the domain to fully discretize the given complex polygonal domain into all quadrilaterals, thus propagating uniform refinement. This simple method generates a mesh whose elements confirm well to the requested shape by refining the problem domain. We have modified these algorithms and demonstrated their use by generating high quality meshes for some typical multiply connected regions: square domains with regular polygonal holes inside and vice versa. We have also made improvements and modifications to to the above triangulation algorithm of the triangle which can now triangulate a trapezium cut out of a triangle. This new algorithm on the triangulation of a trapezium cut out of a triangle is applied to quadrangulate the planar regions in the shape of a circular annulus and square domain with a square hole inside. We have appended MATLAB programs which incorporate the mesh generation schemes developed in this paper. These programs provide valuable output on the nodal coordinates, element connectivity and graphic display of the all quadrilateral mesh for application to finite element analysis.