The following theorem, called Liouville's theorem, is well known. THEOREM 1. Any harmonic function bounded either above or below in all of n-space is constant. The reader is referred to the excellent book by Protter and Weinberger [1] for the proof of the above theorem. It is the purpose of this note to prove an analogous theorem for biharmonic functions and study some applications. Now u is biharmonic in an open set D of the n-dimensional space En if it obeys