Abstract: The finite element method is an important method to solveboundary value problems. In two dimensional problems, the constant strain three-nodetriangular element and the bilinear four-node quadrilateral element arewidely used. Irregular polygonal elements can be used not only toconveniently and effectively simulate mechanical properties of materials,but also to enhance flexibility in meshing. For complexgeometries, the polygonal element grid enjoys greater advantages. In the past decade,researchers have shown interestis in the numerical methods based on polygonalelements, and have obtained some new results. In this paper the advancesinpolygonal finite elements are reviewed. The development of polygonalfinite elements is discussed, including Wachspressinterpolation, Laplace interpolation and barycentric coordinates. Unlikethe polynomial form of shapefunctions in the classical finite element, the shape functions of a polygonalelement can take both rational and irrational forms. The shape functionsinterpolate nodal values, satisfy linear completeness, can be used to reconstructthe linear displacement field, and permit the direct imposition ofessential boundary conditions as in the conventional finite element method. Theyare linear on the boundary of a polygonal element, which ensuresautomatically the consistency of inter-elements. The shape functionshave a uniformformulation for different side number elements, so one can convenientlyprogram for a variety of meshes. Some issues for future development ofpolygonal finite elements are also discussed.