A short and self-contained presentation of the results known about the Pom-peiu problem is given and a new result is obtained. In particular, it is proved that if D 1 has Pompeiu's property (P-property) then D 2 has it, provided D 2 is sufficiently close to D 1 in the following sense: meas(D 12 \ D 12) is sufficiently small. Here D 12 := D 1 ∪ D 2 , D 12 := D 1 ∩ D 2 .