Limit theorems for functionals of stationary Gaussian fields

In this talk, we explore central and non-central limit theorems for functionals of stationary Gaussian random fields, together with their regularity properties, through the lens of their decomposition into Wiener chaoses. We will emphasize three main settings: the case in which the functional is ``concentrated on a single chaos''; the Breuer-Major setting, where all chaotic components contribute equally to the limit; and the challenging scenario in which the dominant contribution to the variance arises from the tail of the chaotic expansion. Throughout the presentation, we will illustrate these phenomena with key examples and discuss some open questions. This talk is mainly based on joint works with L. Gass, L. Maini, N. Turchi and G. Zheng.