We define the concept of unique exchange on a sequence ( X 1 ,…, X m ) of bases of a matroid M as an exchange of x ϵ X i for y ϵ X j such that y is the unique element of X j which may be exchanged for x so that ( X i − { x }) ∪ { y } and ( X j − { y }) ∪ { x } are both bases. Two sequences X and Y are compatible if they are on the same multiset. Let UE(1) [ UE(2) ] denote the class of matroids such that every pair of compatible basis sequences X and Y are related by a sequence of unique exchanges [unique exchanges and permutations in the order of the bases]. We similarly define UE(3) by allowing unique subset exchanges. Then UE(1), UE(2) , and UE(3) are hereditary classes (closed under minors) and are self-dual (closed under orthogonality). UE(1) equals the class of series-parallel networks, and UE(2) and UE(3) are contained in the class of binary matroids. We conjecture that UE(2) contains the class of unimodular matroids, and prove a related partial result for graphic matroids. We also study related classes of matroids satisfying transitive exchange, in order to gain information about excluded minors of UE(2) and UE(3) . A number of unsolved problems are mentioned.