General-purpose, moving-mesh schemes for hydrodynamics have opened the possibility of combining the accuracy of grid-based numerical methods with the flexibility and automatic resolution adaptivity of particle-based methods. Due to their supersonic nature, Keplerian accretion discs are in principle a very attractive system for applying such freely moving-mesh techniques. However, the high degree of symmetry of simple accretion disc models can be difficult to capture accurately by these methods, due to the generation of geometric grid noise and associated numerical diffusion, which is absent in polar grids. To explore these and other issues, in this work we study the idealized problem of two-dimensional planet-disc interaction with the moving-mesh code AREPO. We explore the hydrodynamic evolution of discs with planets through a series of numerical experiments that vary the planet mass, the disc viscosity and the mesh resolution, and compare the resulting surface density, vortensity field and tidal torque with results from the literature.We find that the performance of the moving-mesh code in this problem is in accordance with published results, showing good consistency with grid codes written in polar coordinates. We also conclude that grid noise and mesh distortions do not introduce excessive numerical diffusion. Finally, we show how the moving-mesh approach can help in resolving an outstanding challenge for polar-coordinate grid codes, namely the successful implementation of adaptive mesh refinement in regions of high density around planets and planetary wakes, while retaining the background flow at low resolution.