The structure of a stationary Gaussian process near a local maximum with a prescribed height$u$has been explored in several papers by the present author, see [5]–[7], which include results for moderate$u$as well as for$u$→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ$(t) t ∈ R$^{n}}, with mean zero and the covariance function$r$($t$). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type.