Abstract In this paper we extend some well known properties of monotone and maximal monotone operators to the wider class of e-monotone and maximal e-monotone operators. The main results concern local boundedness of maximal e-monotone operators, maximal 2e-monotonicity of the Clarke–Rockafellar subdifferential $$\partial ^{CR}f$$ ∂ CR f for an e-convex function f, and the characterization of e-monotonicity of an operator T via the behaviour of its e-Fitzpatrick function outside the graph of T.