Variations in the distribution of mass within the atmosphere and changes in the pattern of winds, particularly the strength and location of the major mid-latitude jet-streams, produce fluctuations in all three components of the angular momentum of the atmosphere on timescales upwards of a few days. In a previous study (Hide et al. 1980) it has been shown that variations in the axial component of atmospheric angular momentum during the Special Observing Periods in 1979 of the First GARP Global Experiment (FGGE, where GARP is the Global Atmospheric Research Program) are well correlated with changes in length-of-day. This would be expected if the total angular momentum of the atmosphere and 'solid' Earth were conserved on short timescales (allowing for lunar and solar effects) but not if angular momentum transfer between the Earth's liquid core and solid mantle, which is accepted to be substantial and even dominant on time-scales upwards of several years, were significant on timescales of weeks or months. Fluctuations in the equatorial components of atmospheric angular momentum should contribute to the observed wobble of the instantaneous pole of the Earth's rotation with respect to the Earth's crust, but this has not been shown conclusively by previous studies. In this paper we re-examine some aspects of the underlying theory of non-rigid body rotational dynamics and angular momentum exchange between the atmosphere and solid Earth. Since only viscous or topographic coupling between the atmosphere and solid Earth can transfer angular momentum, no atmospheric flow that everywhere satisfied inviscid equations (including, but not solely, geostrophic flow) could affect the rotation of a spherical solid Earth. Currently available meteorological data are not adequate for evaluating the usual wobble excitation functions accurately, but we show that partial integration leads to an expression involving simpler functions-here called 'equatorial angular momentum functions' - which can be reliably evaluated from available meteorological data. The length-of-day problem is treated in terms of a similar 'axial angular momentum function'; and 'effective angular momentum functions' are defined in order to allow for rotational and surface loading deformation of the Earth. Daily values of these atmospheric angular momentum functions have been calculated from the 'initialized analysis global database' of the European Centre for Medium-Range Weather Forecasts (ECMWF). They are presented for the period 1 January 1981-30 April 1982, along with the corresponding astronomically observed changes in length-of-day and polar motion, published by the Bureau International de l'Heure (BIH). Changes in length-of-day during this period can be accounted for almost entirely by angular momentum exchange between the atmosphere and solid Earth, and the existence of a persistent fluctuation in this exchange, with a time-scale of about 7 weeks, is confirmed. We also demonstrate that meteorological phenomena provide an important contribution to the excitation of polar motion. Our work offers a theoretical basis for future routine determinations of atmospheric angular momentum fluctuations for the purposes of meteorological and geophysical research, including the assessment of the extent to which movements in the solid Earth associated with very large earthquakes contribute to the excitation of the Chandlerian wobble.