With the development of airborne gravimetry and gradiometry, efficient algorithms are needed to compute gravitational effects of the topography. In this paper, such algorithms based on the fast Fourier transform (FFT) are developed and used for studying effects of terrain representation (mass lines or prisms), height data resolution needed, and the number of expansion terms required to approximate the basically nonlinear terrain effect integrals. In addition to airborne gravity and gradiometry, improved formulas are also given for terrestrial gravity, i.e., the classical terrain correction. Results of the method are given for a very rough terrain in the Kananaskis area of the Canadian Rocky Mountains assuming a constant density value of 2.67 g/cm3. Comparisons have been done for height data grid spacings of 0.1, 0.2, 0.5, and 1 km at two different flight altitudes. Results indicate that to obtain gravity or gravity gradient effects at accuracies of 0.3--0.5 mGal and 1 Eotvos (E) unit, respectively, a 0.5 km¿0.5 km grid is needed for a flying altitude of 1 km above the topography. Lowering the flight altitude to 600 m, a typical value for gradiometer surveys, requires height data given at a grid spacing of 0.25 km, as well as the inclusion of third-order terms in the expansions of the terrain correction integrals. The advantage of the FFT method is computational speed: the typical CPU time for computing gravity and all gravity gradient tensor elements on a 114¿74 point grid was about 15 s on a large mainframe (CDC Cyber 175). ¿ American Geophysical Union 1988