In this talk, we develop and analyze an arbitrary Lagrangian--Eulerian discontinuous Galerkin (ALE-DG) method for solving one-dimensional hyperbolic equations involving δ-singularities on moving meshes. The 𝐿2 and negative norm error estimates are proven for the ALE-DG approximation. More precisely, when choosing the approximation space with piecewise 𝑘th degree polynomials, the convergence rate in 𝐿2-norm for the scheme with the upwind numerical flux is (𝑘+1)th order in the region apart from the singularities, the convergence rate in 𝐻−(𝑘+1) norm for the scheme with the monotone fluxes in the whole domain is 𝑘th order, the convergence rate in 𝐻−(𝑘+2)norm for the scheme with the upwind flux in the whole domain can achieve (𝑘+1/2)th order, and the convergence rate in 𝐻−(𝑘+1)(𝑅\𝑅𝑇) norm for the scheme with the upwind flux is (2𝑘+1)th order, where 𝑅𝑇 is the pollution region at time 𝑇 due to the singularities. Moreover, numerically the (2𝑘+1)th order accuracy for the postprocessed solution in the smooth region can be obtained, which is produced by convolving the ALE-DG solution with a suitable kernel consisting of B-splines. Numerical examples are shown to demonstrate the accuracy and capability of the ALE-DG method for the hyperbolic equations involving 𝛿-singularity on moving meshes.