The periodically forced Duffing oscillator is a well-known, earliest example of dynamical systems exhibiting chaotic dynamics. For example, such behavior was reported to be observed in analog simulations in 1960's. However, the occurrence of chaotic dynamics has been mathematically proven only for limiting cases. In this talk, we consider the Duffing equation with a parametric forcing and prove that it is generally (meromorphically) nonintegarable in the meaning of Bogoyavlenski after reviewing a mathematical proof for the existence of chaos in some case by Melnikov's method. Our main tools are an extension of Morales-Ramis theory due to Ayoul and Zung, and Kovacic algorithm.