Abstract The sequence A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be written as the sum of h elements of A. Let M h A denote the set of elements that have more than one representation as a sum of h elements of A. It is proved that there exists an asymptotic basis A such that M h A(x) = O(x 1−1 h+ϵ ) for every ϵ > 0. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It is proved that there does not exist a sequence A that is simultaneously a minimal basis of orders 2, 3, and 4. Several open problems concerning minimal bases are also discussed.