Abstract. A positive de nite Hermitian lattice is said to be 2-universalif it represents all positive de nite binary Hermitian lattices. We ndsome niteness theorems which ensure 2-universality of Hermitian latticesover several imaginary quadratic number elds. 1. IntroductionSince the celebrated Lagrange’s Four Square Theorem, an important sub-ject is positive de nite quadratic forms which represent all positive integers.This subject was studied by many mathematicians, for instance Jacobi, Pepin,Liouville, etc. In particular, Ramanujan found all 54 positive de nite integralquaternary diagonal forms represent all positive integers. Dickson called suchquadratic forms universal.In 1948, Willerding investigated the universal classically integral quaternaryforms and found 178 such forms in her dissertation [11]. In her dissertationshe had to check as many as 1046 quaratic forms and the methods are toocomplicated to be veri ed. Indeed, her list contains some mistakes, but thecorrection was performed in the di erent directions.In 1993, Conway and Schneeberger proved the so-called Fifteen Theoremwhich ensures the universality of positive de nite classically integral quadraticforms. It states that if a positive de nite quadratic form represents 1, 2, 3, 5,6, 7, 10, 14, and 15, then it represents all positive integers. This astoundingtheorem enables them to check Willerding’s list. Consequently, they obtain 204positive de nite universal quaternary classically integral quadratic forms.After several years, Bhargava proposed a simpler and more elegant proofthan Conway and Schneeberger’s original one. Besides he insisted that forevery in nite subset S of N, there is a nite subset S