The use of more than one cell topology in unstructured meshes may impose additional limitations upon the intrinsic adaptivity of the mesh. Additional geometrical entities have been used to install descriptions of arbitrary polyhedra within an unstructured mesh, permitting the incorporation of any mix of cell topologies, whilst maintaining the intrinsic of the mesh. Three-dimensional, unstructured, hybrid meshes are constructed using a modified multi-block approach in which unstructured data is used to naturally incorporate mesh singularities, and permit zonal adaptations. The meshes are used to demonstrate a finite volume Euler flow solver, which efficiently operates on an edge structure such that no limitations are imposed upon the mesh topology. Results obtained using a first order upwind scheme demonstrate the shock capturing abilities for a range of two and three-dimensional transonic flows. The implementation of a higher order method, using the MUSCL formulae, illustrates the non-trivial nature of applying such techniques within an arbitrary cell environment, whilst solutions obtained for two and three-dimensional transonic flows demonstrate the increased resolution obtained. The implementation across distributed platforms is described in detail, with good performance results presented for a range of architectures, including a workstation cluster and an IBM SP2. An adaptive mesh algorithm is employed to automatically identify and resolve local flow features. No limitations are placed on the adaptation of mesh cells, which is demonstrated for a supersonic internal channel flow, where the strong shock waves are clearly captured using a hexahedral to polyhedral strategy.