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Some psychoanalysts and neuroscientists assert that mathematical ways of thinking, variously called nonlinear dynamics, complexity theory, chaos theory, catastrophe theory, small-world theory, and artificial life (to name a few), hold keys that integrate and enrich the study of human minds, feeling, and brain function. They refer to ideas like nonlinearity, complexity, chaos, the fractal nature of reality, and emergence, suggesting that puzzling aspects of mental life can be comprehended using them. Is there content behind these obscure words? Can mathematics contribute to psychoanalysis? Does this project continue the intellectual imperialism of the “hard sciences” over psychology? How can an analyst come to a position of understanding and rationally evaluate these claims or even learn enough to appropriately use these ideas? In this review I will try to provide a road map for analysts to begin an intellectually credible exploration of nonlinear dynamics. I have deliberately omitted any discussion of specific applications of such dynamics to psychoanalysis.1 My purpose is to help you find your way into them from a sound base in the underlying theories.