<p>Recently, two extraordinary results on aperiodic monotiles have been obtained in two different settings. One is a family of aperiodic monotiles in the plane discovered by Smith et al. (2024), where rotation is allowed, breaking the 50-year-old record (aperiodic sets of two tiles found by Penrose in the 1970s) on the minimum size of aperiodic sets in the plane. The other is the existence of an aperiodic monotile in the translational tiling of ℤ^<i>n</i> for some huge dimension <i>n</i>, proved by Greenfeld and Tao (2024). This disproves the long-standing periodic tiling conjecture. However, it is known that there is no aperiodic monotile for translational tiling of the plane. The smallest size of known aperiodic sets for translational tilings of the plane is 8, which was discovered more than 30 years ago by Ammann (1992). In this paper, we prove that there exists a translational aperiodic set of 7 polyominoes. This breaks the 30-year-old record of Ammann (1992). In addition to the aperiodicity, we also show that the translational tiling of the plane with a set of 7 polyominoes is undecidable.</p>