The mean integral of harmonic functions on balls centered at x equals the value of these functions at x. This is the well known Gauss mean value theorem. In 1972 Kuran proved the reverse: if D is a bounded open set containing x, such that the mean integral of harmonic functions on D equals the value of these functions at x, then D is a ball centered at x. Two questions may be raised: (1) similar rigidity results can be proved for weighted mean integrals? (2) is the Gauss mean value formula stable? That is: if the mean integral of harmonic functions on D centered at x is almost equal to the value of these functions at x, then D is almost a ball with center x? In this talk I will discuss recent results on these issues obtained in collaboration with E. Lanconelli (1) and with N. Fusco, E. Lanconelli and X. Zhong (2)