In order to improve the Severi inequality,Barja,Pardini and Stoppino defined the so-called eventual paracanonical maps of varieties of maximal Albanese dimensions.They proved that the degrees of these maps of surfaces are bounded by 4.We shall give a complete description of the eventual paracanonical maps of surfaces and give an optimal bound of the degrees of these maps of threefolds.This is a joint work with Zhiyu Tian.