  A positive definite integral quadratic form is said to be almost
(primitively) universal if it (primitively) represents all but at most finitely
many positive integers. In general, almost primitive universality is a stronger
property than almost universality. The two main results of this paper are: 1)
every primitively universal form nontrivially represents zero over every ring
of p-adic integers, and 2) every almost universal form in five or more
variables is almost primitively universal.
