We propose an Eulerian-Lagrangian (EL) Runge-Kutta (RK) discontinuous Galerkin (DG) method. The method is designed as a generalization of the semi-Lagrangian (SL) DG method, which is formulated based on an adjoint problem and tracing upstream cells by tracking characteristics curves highly accurately. Depending on the velocity field, the shape of upstream cells could be of arbitrary shape, for which a more sophisticated approximation is required to get high order approximation. For example, in the SLDG algorithm, quadratic-curved (QC) quadrilaterals were proposed to approximate upstream cells in order to obtain third order spatial accuracy in a swirling deformation example. In this work, we propose a more general formulation, named the ELDG method, for which the scheme is based on a modified adjoint problem for which upstream cells are always quadrilaterals. This leads to a new formulation of ELDG method, which avoids the need to use QC quadrilaterals to better approximate upstream cells in the original SLDG algorithm. The newly proposed ELDG method can be viewed as a new general framework, in which both the classical RK DG formulation and the SL DG formulation can fit in. Numerical results on linear advection problems, as well as the nonlinear Vlasov dynamics using the exponential RK framework, will be presented to demonstrate the effectiveness of the proposed approach.