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We describe various approaches to understanding Fukaya categories ofcotangent bundles. All of the approaches rely on introducing a suitable classof noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow(math/0604379, math/0612399) and the authors (math/0701783), before discussinga new approach using family Floer cohomology and the ``wrapped Fukayacategory''. The latter, inspired by Viterbo's symplectic homology, emphasisesthe connection to loop spaces, hence seems particularly suitable when trying toextend the existing theory beyond the simply-connected case.

The symplectic geometry of cotangent bundles from a categorical viewpoint