This paper is a survey of the upper bounds on the complexity of basic algebraic and geometric operations with Pfaffian and Noetherian functions, and with sets definable by these functions. Among other results, we consider bounds on Betti numbers of sub-Pfaffian sets, multiplicities of Pfaffian intersections, effective Lojasiewicz inequality for Pfaffian functions, computing frontier and closure of restricted semi-Pfaffian sets, constructing smooth stratifications and cylindrical cell

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... ons (including an effective version of the complement theorem for restricted sub-Pfaffian sets), relative closures of non-restricted semi-Pfaffian sets and bounds on the number of their connected components, bounds on multiplicities of isolated solutions of systems of Noetherian equations.