The theory of measures of noncompactness has many applications in Topology, Functional analysis and Operator theory. There are many nonequivalent definitions of this notion on metric and topological spaces. First of them was introduced by Kuratowski in 1930. In this paper we consider one axiomatic approach to this notion which includes the most important classical definitions. We also give some new properties of the inner Hausdorff measure of noncompactness on metric linear spaces.