It very much depends on the application. Econometrics uses lots of statistics. Finance uses stochastic calculus. Microeconomics uses linear algebra and calculus. To see how incomprehensible things can get, here's a paper by one of my old econometrics professors: http://econweb.ucsd.edu/~bbeare/pdfs/vinecopula.pdf

Financial economics literature is quite heavy on stochastic processes. This requires a sophisticated treatment of probability, which in turn necessitates familiarity with measure theory. Throw some stochastic differential equations into the mix and you're probably also going to need some background in functional analysis. Not so familiar with the rest of the econ literature, but even basic econometrics requires familiarity with point set topology.

Game theory