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In this article we present the first better than second order accurateunstructured Lagrangian-type one-step WENO finite volume scheme for thesolution of hyperbolic partial differential equations with non-conservativeproducts. The method achieves high order of accuracy in space together withessentially non-oscillatory behavior using a nonlinear WENO reconstructionoperator on unstructured triangular meshes. High order accuracy in time isobtained via a local Lagrangian space-time Galerkin predictor method thatevolves the spatial reconstruction polynomials in time within each element. Thefinal one-step finite volume scheme is derived by integration over a movingspace-time control volume, where the non-conservative products are treated by apath-conservative approach that defines the jump terms on the elementboundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian(ALE) method, where the mesh velocity can be chosen independently of the fluidvelocity.  The new scheme is applied to the full seven-equation Baer-Nunziato model ofcompressible multi-phase flows in two space dimensions. The use of a Lagrangianapproach allows an excellent resolution of the solid contact and the resolutionof jumps in the volume fraction. The high order of accuracy of the scheme inspace and time is confirmed via a numerical convergence study. Finally, theproposed method is also applied to a reduced version of the compressibleBaer-Nunziato model for the simulation of free surface water waves in movingdomains. In particular, the phenomenon of sloshing is studied in a moving watertank and comparisons with experimental data are provided.

High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes  for Non-Conservative Hyperbolic Systems: Applications to Compressible  Multi-Phase Flows