Abstract. The power of atmospheric circulation is a key measure of the Earth's climate system. The mismatch between predictions and observations under a warming climate calls for a reassessment of how atmospheric power W is defined, estimated and constrained. Here we review published formulations for W and show how they differ when applied to a moist atmosphere. Three factors, a non-zero source/sink in the continuity equation, the difference between velocities of gaseous air and condensate, and interaction between the gas and condensate modyfing the equations of motion, affect the formulation of W. Starting from the thermodynamic definition of mechanical work, we derive an expression for W from an explicit consideration of the equations of motion and continuity. Our analyses clarify how some past formulations are incomplete or invalid. Three caveats are identified. First, W critically depends on the boundary condition for gaseous air velocity at the Earth's surface. Second, confusion between gaseous air velocity and mean velocity of air and condensate in the expression for W results in gross errors despite the observed magnitudes of these velocities are very close. Third, W expressed in terms of measurable atmospheric parameters, air pressure and velocity, is scale-specific; this must be taken into account when adding contributions to W from different processes. We further present a formulation of the atmospheric power budget, which distinguishes three components of W: the kinetic power associated with horizontal pressure gradients (W_K), the gravitational power of precipitation (W_P) and the condensate loading (W_c). This formulation is valid with an accuracy of the squared ratio of the vertical to horizontal air velocities. Unlike previous approaches, it allows evaluation of W_P + W_c without knowledge of atmospheric moisture or precipitation. This formulation also highlights that W_P and W_c are the least certain terms in the power budget as they depend on vertical velocity; W_K depending on horizontal velocity is more robust. We use MERRA and NCAR/NCEP re-analyses to evaluate the atmospheric power budget at different scales. Estimates of W_K are found to be consistent across the re-analyses, while estimates for W and W_P drastically differ. We then estimate independent precipitation-based values of W_P and discuss how such estimates could reduce uncertainties. Our analyses indicate that W_K increases with temporal resolution approaching our theoretical estimate for condensation-induced circulation when all convective motion is resolved. Implications of these findings for constraining global atmospheric power are discussed.