The present thesis considers the development and analysis of arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods with time-dependent approximation spaces for conservation laws and the Hamilton-Jacobi equations. Fundamentals about conservation laws, Hamilton-Jacobi equations and discontinuous Galerkin methods are presented. In particular, issues in the development of discontinuous Galerkin (DG) methods for the Hamilton-Jacobi equations are discussed. The development of the ALE-DG methods based on the assumption that the distribution of the grid points is explicitly given for an upcoming time level. This assumption allows to construct a time-dependent local affine linear mapping to a reference cell and a time-dependent finite element test function space. In addition, a version of Reynolds’ transport theorem can be proven. For the fully-discrete ALE-DG method for nonlinear scalar conservation laws the geometric conservation law and a local maximum principle are proven. Furthermore, conditions for slope limiters are stated. These conditions ensure the total variation stability of the method. In addition, entropy stability is discussed. For the corresponding semi-discrete ALE-DG method, error estimates are proven. If a piecewise $\mathcal{P}^{k}$ polynomial approximation space is used on the reference cell, the sub-optimal $\left(k+\frac{1}{2}\right)$ convergence for monotone fuxes and the optimal $(k+1)$ convergence for an upwind flux are proven in the $\mathrm{L}^{2}$-norm. The capability of the method is shown by numerical examples for nonlinear conservation laws. Likewise, for the semi-discrete ALE-DG method for nonlinear Hamilton-Jacobi equations, error estimates are proven. In the one dimensional case the optimal $\left(k+1\right)$ convergence and in the two dimensional case the sub-optimal $\left(k+\frac{1}{2}\right)$ convergence are proven in the $\mathrm{L}^{2}$-norm, if a piecewise $\mathcal{P}^{k}$ polynomial approximation space is used on the reference cell. For the fullydiscrete method, the geometric conservation is proven and for the piecewise constant forward Euler step the convergence of the method to the unique physical relevant solution is discussed.