The lecture is dedicated to the study of Gaussian random fields through their geometrical properties. In the first section, we introduce general definitions and invariance properties of Gaussian fields indexed by the Euclidean space \(\mathbb {R}^d\) . In particular, the crucial role of the covariance function is emphasized and the special case of stationary fields is presented from the Gaussian random wave point of view. The second section deals with a geometric feature that is really specific to the multivariate context: anisotropy. We present various models of Gaussian fields whose distributions all share the property of not being invariant under rotations. We try to understand which characteristics of the field are impacted by the anisotropy property and how it affects them. In the third section, we provide Kac-Rice formulas and the so-called Gaussian kinematic formulas. They consist in explicitly writing moments of some geometric functionals that depend on the level sets of the Gaussian field. These formulas are perfect tools to study both qualitative and quantitative geometric properties of stationary Gaussian fields. This is done in the last section where we visit recent works that aim at inferring features of the considered Gaussian field based on sparse geometrical observations.