Sarnak and his collaborators initiated a program to investigate the distribution of points whose coordinates have few prime factors on varieties equipped with a group structure. In this talk, we shall concentrate on the case of unirational varieties. We prove that there exists an integer $r$ such that rational points for which the product of the coordinates has at most $r$ prime factors form a Zariski dense subset, provided that the unirational variety has one rational point. Moreover, we may obtain rather small bounds for $r$ for several special cases. The proof relies on the geometric structure of the varieties and the use of analytic tools.