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We review several properties of integrals of the Wigner distribution onsubsets of the phase space. Along our way, we provide a theoretical proof ofthe invalidity of Flandrin's conjecture, a fact already proven via numericalarguments in our joint paper [MR4054880] with B.Delourme and T.Duyckaerts. Weuse also the J.G.Wood and A.J.Bracken paper [MR2131219], for which we offer amathematical perspective. We review thoroughly the case of subsets of the planewhose boundary is a conic curve and show that Mehler's formula can be helpfulin the analysis of these cases, including for the higher dimensional caseinvestigated in the paper [MR2761287] by E.Lieb and Y.Ostrover. Using theFeichtinger algebra, we show that, generically in the Baire sense, the Wignerdistribution of a pulse in $L^2(\mathbb R^n)$ does not belong to $L^1(\mathbbR^{2n})$, providing as a byproduct a large class of examples of subsets of thephase space $\mathbb R^{2n}$ on which the integral of the Wigner distributionis infinite. We study as well the case of convex polygons of the plane, with arather weak estimate depending on the number of vertices, but independent ofthe area of the polygon.

Integrating the Wigner Distribution on subsets of the phase space, a Survey