A new class of finite elements is described for dealing with non-matching meshes, for which the existing finite elements are hardly efficient. The approach is to employ the moving least-square (MLS) scheme to devise a class of elements with an arbitrary number of nodal points on the parental domain. This approach generally leads to elements with the rational shape functions, which significantly extends the function space of the conventional finite element method. With a special choice of the nodal points and the base functions, the method results in useful elements with the polynomial shape functions for which the C-1 continuity breaks down across the boundaries between the subdomains comprising one element. The present scheme possesses an extremely high potential for applications which deal with various problems with discontinuities, such as material inhomogeneity, crack propagation, phase transition and contact mechanics. The effectiveness of the new elements for handling the discontinuities due to non-matching interfaces is demonstrated using appropriate examples. Copyright (c) 2007 John Wiley & Sons, Ltd.