We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. The notion is named b-minimality and is based on definable families of points and balls. We develop a dimension theory and prove a cell decomposition theorem for b-minimal structures. We show that b-minimality applies to the theory of Henselian valued fields of characteristic zero, generalizing work by Denef–Pas [25, 26]. Structures which are o-minimal, v-minimal, or p-minimal and which satisfy some slight extra conditions are (...) also b-minimal, but b-minimality leaves more room for nontrivial expansions. The b-minimal setting is intended to be a natural framework for the construction of Euler characteristics and motivic or p-adic integrals. The b-minimal cell decomposition is a generalization of concepts of Cohen [11], Denef [15], and the link between cell decomposition and integration was first made by Denef [13]. (shrink)