Important classes of locally compact groups are characterized by their actions by affine isometries on Hilbert spaces (groups with Kazhdan's property, a-T-menable groups aka groups with the Haagerup property). We will be interested on the question of irreducibility of such actions, in the sense that the only non empty closed invariant affine subspace is the whole space. This notion was extensively studied in a recent joint work of T. Pillon, A. Valette and myself. We will report on this work as well as on some further results. Special attention will be paid to affine actions whose linear part is a factorial representation, that is, a representation which generates a factor in the von Neumann algebra sense

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