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Starting from the BGK model of the Boltzmann equation, we develop a robust and accurate finite volume gas kinetic scheme on unstructured triangular meshes. The proposed numerical approach is composed of two steps—an initial reconstruction step and a gas evolution step. In the initial reconstruction step, an unstructured version of the local extremum diminishing interpolation is applied to the conservative variables and to compute left and right states along a node edge. In the gas evolution step, the local integral solution of the BGK model is used to compute numerical fluxes at a cell interface. This approach provides an alternative to Riemann solvers and yields numerical schemes which possess many desirable properties that may not be found in Godunov-type schemes. A classich-refinement adaptive procedure is implemented to increase the spatial resolution of high-speed unsteady flow characteristics such as shock waves, contact discontinuities, or expansion waves with minimal computational costs and memory overheads. This procedure involves mesh enrichment/coarsening steps to either insert nodes on an edge center in high-gradient regions or delete nodes in over-resolved regions. Numerical results of several test cases for unsteady compressible inviscid flows are presented. In order to verify the accuracy and robustness of the current numerical approach, the computed results are compared with analytical solutions, experimental data, the results of structured mesh calculations, and the results obtained by widely used flux splitting methods.