This report documents the potential capabilities of adaptive inviscid flow calculations on unstructured meshes in three dimensions using the finite element method. The finite element formulation of the compressible Euler and Navier-Stokes equations is based on a two-step explicit Taylor-Galerkin scheme. Adaptive remeshing is applied to enhance the numerical solution in the vicinity the shocks. Particular emphasis is put on the generation of unstructured tetrahedral meshes as well as in the discussion of estimating the error in the numerical solution. Finally, an application to a well-known 3D high speed flow problem is shown where adaptive remeshing techniques become essential to keep the computational cost within reasonable limits. Its results are compared to some seemingly best reference solutions using other algorithms.