<p>We propose and study the class of monotonic mean–deviation risk measures, represented by a combination of a risk-weighted deviation functional and the expectation. These risk measures belong to the class of consistent risk measures and admit an axiomatic characterisation via preference relations. By further assuming the convexity and linearity of the risk-weighting function, we obtain convex and coherent risk measures within this class, giving rise to many new explicit examples of convex and nonconvex consistent risk measures. In particular, we specialise to the convex case of the monotonic mean-deviation measure and obtain its dual representation. Further, we establish asymptotic consistency and normality of the natural estimators of the monotonic mean–deviation measures. Finally, monotonic mean–deviation measures are applied to the problem of portfolio selection using financial data.</p>