This paper develops a unified framework for the robustification of risk measures beyond the classical convex and cash-additive setting. We consider general monotone risk measures on L^p spaces and construct their robust counterparts through families of uncertainty sets that capture model ambiguity. Two complementary mechanisms generate quasi-convex robustified measures: one where quasi-convexity is inherited from the initial risk measure under convex uncertainty sets, and another where it stems from the quasi-convex or c-quasi-convex structure of the uncertainty sets themselves. Building on Cerreia-Vioglio et al. (2011); Frittelli and Maggis (2011), we derive dual (penalty-type) representations for robust quasi-convex and cash-subadditive risk measures, showing that the classical convex cash-additive case arises as a special instance. We further analyze acceptance families and capital allocation rules under robustification, highlighting how model uncertainty affects acceptability and the distribution of capital.