Dunstan et al. first proposed the concept of supermatroids in 1972 by generalizing the underlying sets of matroids from finite sets to finite posets. Barnabei et al. introduced another matroidal structure on posets, i.e., poset matroids. By the one-to-one correspondence between finite distributive lattices and finite posets, poset matroids are just supermatroids on distributive lattices. This paper studies axiom systems of supermatroids. Independence axioms of modular supermatroids are proposed, middle base axiom and base exchange axiom of modular supermatroids are proved and modular supermatroids are characterized by these two properties. At last, we point out a mistake made by Barnabei et al. in their circuit axioms of distributive supermatroids and correct this mistake by proposing a new elimination property of circuit, then the circuit axiom of distributive supermatroids is established. As an application of the elimination property, it is proved that an element covering a base in a distributive supermatroid contains only one circtuit.