Which areas of mathematical knowledge are required for understanding different genres of theoretical economics literature (and at which level)? I would like to know which mathematical skills are most important to understanding different types of theoretical economics literature. An ideal answer would list disciplines used for different 'genres' of literature; for example: "Mathematical finance requires elementary linear algebra and stochastic calculus" (is this accurate?).
For microeconomic theory, a starting point would be


Basic multivariate calculus,
topology/set theory
real analysis,
probability theory,
a basic knowledge of linear algebra.


For 'applied' micro theory (theoretical industrial organisation, applications in information economics, etc.), or for reading of a graduate-level microeconomics text (such as Mas-Colell, Whinston, and Green's "Microeconomic Thoery"), a good understanding of the basics of these areas will suffice. Most material will be set in something like a Euclidean space and will draw on a fairly restricted set of methods and results from the above domains. You can already get a long way with a good understanding of the basic continuity properties of the reals ($\epsilon$-$\delta$ proofs, open and closed intervals, intermediate value theorem, basic fixed-point theorems, etc.), the ability to do partial differentiation, integration, and to calculate conditional probabilities and expectations.

As one turns to the journal literature and looks more towards pure (rather than applied) economic theory, the level of mathematical sophistication increases somewhat and it becomes more likely that you will encounter more exotic (from the point of view of economics) mathematical methods and results.