Complex economic models often lack the structure for the application of standard techniques in monotone comparative statics. Generalized monotonicity analysis (GMA) extends the available methods in several directions. First, it provides a way of finding parameter moves that yield monotonicity of model solutions. Second, it allows studying the monotonicity of functions or subsets of variables. Third, GMA naturally provides bounds on the sensitivity of variables to parameter changes. Fourth, GMA may be used to derive conditions under which monotonicity obtains with respect to functions of parameters, corresponding to imposed parameter moves. Fifth, GMA contributes insights into the theory of comparative statics, for example, with respect to dealing with constraints or exploiting additional information about the model structure. Several applications of GMA are presented, including constrained optimization, nonsupermodular games, aggregation, robust inference, and monotone comparative dynamics.