Abstract: By comparing the compact finite difference schemes and discontinuousGalerkin (DG) methods, the concepts of ``static re-construction'' and ``dynamicre-construction'' are proposed for high-order numerical schemes. Based on thenew concept of ``hybrid re-construction'', a novel class of DG/finite volumehybrid schemes (DG/FV schemes) is presented. In our DG/FV schemes, thelower-order derivatives are computed locally in a cell by traditional DGschemes (called as ``dynamic re-construction''), while the higher-orderderivatives are constructed by the ``static re-construction'' of finite volumeschemes, using the known lower-order derivatives in the cell itself and inthe neighbor cells. The DG/FV hybrid schemes can reduce the CPU time andstorage memory greatly than the traditional DG schemes with the same orderof accuracy, and can be extended directly for unstructured and hybrid gridsas the DG and/or FV methods. The DG/FV hybrid schemes are applied for1D and 2D scalar conservation law. The numerical results demonstrate the accuracy,the efficiency, and the super-convergence property in our third-order DG/FVhybrid schemes.