Apositive definite integral quadratic form f is called n-regular if f represents every quadratic form of rank n that is represented by the genus

of f . In this paper, we show that for any integer n greater than or equal to 27, every n-regular (even) form f is (even) n-universal, that is, f represents all (even, respectively) positive definite integral quadratic forms of rank n. As an application, we show that the minimal rank of n-regular forms has an

exponential lower bound for n as it increases.