Large-Eddy Simulation is expected to become a powerful tool for turbulent flow calculations of practical interest in the near future. The paper is concerned with the development and first applications of a finite-volume method based on curvilinear body-fitted grids suitable to simulate flows in or around complex geometries. Two different subgrid-scale models are implemented and tested. Because the near-wall region cannot be resolved for high-Reynolds-number flows, two different wall function approaches are used. The method is applied to various turbulent flow problems e.g. the flow through a straight square duct, the flow around a surface-mounted cubical obstacle, and the flow around a square cylinder. The computed results are compared with experimental data available.