The Lagrangian method is widely used in many fields for multi-material flow simulations due to its distinguished advantage in capturing material interfaces automatically. In this talk, we present our recent research on the design of a class of Lagrangian schemes for solving the compressible Euler equations both on the quadrilateral meshes and curved quadrilateral meshes. The schemes are based on high order essentially non-oscillatory (ENO) and weighted ENO (WENO) reconstruction. They are conservative for the mass, momentum and total energy, can achieve at least uniformly second order accuracy both in space and time with the quadrilateral meshes and uniformly third order accuracy with the curved quadrilateral meshes, are essentially non-oscillatory, and have no parameters to be tuned for individual test cases. One and two dimensional numerical examples are presented to demonstrate the performance of the schemes in terms of accuracy, resolution for discontinuities, and non-oscillatory properties. This is a joint work with Chi-Wang Shu.