We deal with a stationary isotropic random field X:R^d→R and we assume it is partially observed through some level functionals. We aim at providing a methodology for a test of Gaussianity based on this information. More precisely, the level functionals are given by the Euler characteristic of the excursion sets above a finite number of levels. On the one hand, we study the properties of these level functionals under the hypothesis that the random field X is Gaussian. In particular, we focus on the mapping that associates to any level u the expected Euler characteristic of the excursion set above level u. On the other hand, we study the same level functionals under alternative distributions of X, such as chi-square, harmonic oscillator and shot noise. In order to validate our methodology, a part of the work consists in numerical experimentations. We generate Monte-Carlo samples of Gaussian and non-Gaussian random fields and compare, from a statistical point of view, their level functionals. Goodness-of-fit p−values are displayed for both cases. Simulations are performed in one dimensional case (d=1) and in two dimensional case (d=2), using R.