The talk concerns the numerical solution of systems of general nonlinear hyperbolic conservation laws over unstructured grids by means of the residual distribution method. The fundamental problem related to the conservation of the convective fluxes is recalled in the context of monotone shock capturing schemes operating on the quasilinear form of the governing equations. A new formulation of the first order monotone N scheme as an essential building block of higher order non-oscillatory schemes is introduced, relying on the contour integration of the convective fluxes over the boundaries of an element. Its robustness and monotonicity are demonstrated on a number of test problems and the possible fields of applications are discussed. The method has a strong potential in the solution of complex nonlinear conservation laws without the existence of a conservative linearization, such as the ideal magnetohydrodynamics equations or the hydrodynamics equations for real gases. Moreover, the basic concepts are beneficial for the extension of the monotone residual distribution schemes over nonlinear finite elements, e.g. quadrilaterals, hexahedra or cubic elements over triangles.