The design of an antenna calls for definite amplitudes and phases of the currents, but when the antenna has been constructed and adjusted, there will be departures from the design currents because of several factors. The customary procedure of taking radiation patterns and making the final adjustments semi-empirically has usually been satisfactory, but two difficulties have been setting in with the trend towards large antennas of high gain. First, it is impossible to measure the radiation pattern of the largest existing antennas; even the determination of single sections through the pattern or the gain in one direction presents difficulty. Second, the adjustments themselves are more laborious on larger antennas. It is therefore very desirable that the theory of antenna tolerances should be pursued so that the effect of departures can be taken into account, statistically or otherwise, during the design. This paper considers the effects of systematic and random errors on the radiation pattern of antennas representable by a field distribution over an aperture, such as paraboloidal reflectors and large arrays of small elements. In the case of paraboloids, the deterioration in directivity is found to depend on the mean square departure of the surface from the paraboloid of best weighted least-squares fit and on the two-dimensional autocorrelation function of the departure. The variation of directivity with wavelength of a particular paraboloid is deduced by leaving out of account those two-dimensional Fourier components of the departure with spatial periods less than a wavelength. Practical steps are considered for unifying testing, adjusting, and design so as to lead to the greatest relaxation of the mechanical tolerances imposed on construction.