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We develop a class of Lagrangian schemes for solving the Euler equations of compressible gas dynamics 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 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. [ABSTRACT FROM AUTHOR]