An adaptive polygon scaled boundary finite element method (APSBFEM) is developed for elastodynamic problems. Flexible polygon meshes are generated from background Delaunay triangular meshes and used to calculate structure’s dynamic responses. In each time step, a posteriori-type energy error estimator is employed to locate the polygon subdomains with exceeding spatial discretization error, then edge midpoints of the corresponding triangles are inserted into the background. A new Delaunay triangular mesh and a polygon mesh are regenerated successively. The state variables, including displacement, velocity and acceleration are mapped from the old polygon mesh to the new one by a simple algorithm. A benchmark elastodynamic problem is modeled to validate the developed method. The results show that the adaptive meshes are capable of capturing the steep stress gradient areas, and the dynamic responses agree well with those from the adaptive finite element method and the general polygon scaled boundary finite element method using fine meshes.