For numerical weather prediction with primitive equations (the Eulerian hydrodynamic equations modified by the assumption of hydrostatic equilibrium), various coordinate systems are used to represent the vertical structure of the atmosphere. In this paper, we review the essential features of prediction equations, satisfying the conservation of mass and total energy, in various vertical coordinate systems. We formulate the equations of horizontal motion, hydrostatic balance, mass continuity, and thermodynamics using a generalized vertical coordinate in which any variable that gives a single-valued monotonic relationship with a geometric height can be used as a vertical coordinate. Conditions to conserve total energy in a generalized vertical coordinate are investigated. Various prediction schemes using pressure, height, and potential temperature as a vertical coordinate are derived from the set of basic equations in the generalized coordinate system. These three coordinate systems are unique in th... Abstract For numerical weather prediction with primitive equations (the Eulerian hydrodynamic equations modified by the assumption of hydrostatic equilibrium), various coordinate systems are used to represent the vertical structure of the atmosphere. In this paper, we review the essential features of prediction equations, satisfying the conservation of mass and total energy, in various vertical coordinate systems. We formulate the equations of horizontal motion, hydrostatic balance, mass continuity, and thermodynamics using a generalized vertical coordinate in which any variable that gives a single-valued monotonic relationship with a geometric height can be used as a vertical coordinate. Conditions to conserve total energy in a generalized vertical coordinate are investigated. Various prediction schemes using pressure, height, and potential temperature as a vertical coordinate are derived from the set of basic equations in the generalized coordinate system. These three coordinate systems are unique in th...