We establish the first axiomatic theory for diversification indices using six intuitive axioms -- non-negativity, location invariance, scale invariance, rationality, normalization, and continuity -- together with risk measures. The unique class of indices satisfying these axioms, called the diversification quotients (DQs), are defined based on a parametric family of risk measures. DQ has many attractive properties, and it can address several theoretical and practical limitations of existing indices. In particular, for the popular risk measures Value-at-Risk and Expected Shortfall, DQ admits simple formulas, it is efficient to optimize in portfolio selection, and it can properly capture tail heaviness and common shocks which are neglected by traditional diversification indices. When illustrated with financial data, DQ is intuitive to interpret, and its performance is competitive against other diversification indices.