In this report, we have designed an Essentially Non Oscillatory reconstruction for functions defined on finite-element type meshes. Two related problems are studied: The interpolation of possibility unsmooth multivariate functions on arbitrary meshes and the reconstruction of a function from its averages in the control volumes surrounding the nodes of the mesh. Concerning the first problem, we have studied the behaviour of the highest coefficients of two polynomials interpolations of a function that may admit discontinuities of locally regular curves: the Lagrange interpolation and an approximation such that the mean of the polynomial on any control volume is equal to that of the function to be approximated. This enables us to choose the best stencil for the approximation. The choice of the smallest possible number of stencils is addressed. Concerning the reconstruction problem, two methods have been studied: a first one based on an adaptation of the so-called reconstruction via deconvolution method to irregular meshes and a second one that lies on the approximation on the mean as defined above. The first method is conservative up to a quadrature formula and the second one is exactly conservative. The two methods have the expected order of accuracy, but the second one is much less expensive than the first one. Some numerical examples are given which demonstrate the efficiency of the reconstruction.