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Morse theory, along with its intimidating infinite dimensional cousin discovered by Floer, has played a fundamental role in developments across topology in the last 50 years. When first developing his theory for applications to symplectic topology, Floer noticed that Morse theory had hitherto neglected much of the information that is encoded in the spaces of solutions to PDE's that underlie its construction, by focusing attention only on those spaces of solutions which consist of points and 1-manifolds, at the expense of those of higher dimension, and he foresaw that this additional information would allow one to extract generalized homology groups (such as K-theory or cobordism) from Morse theory. In this series of lectures, I will begin by introducing Floer's insight as an outgrowth of basic ideas of the subject, going back to Pontryagin and Thom, which express the space manifolds up to cobordism in linear algebraic terms (Grassmannians). I will then explain how recent technical advances have realized Floer's vision, and have yielded concrete applications to the topology of symplectic manifolds. Finally, I will describe work in progress for understanding the totality of the structure that appears in Floer's theory, yielding a reformulation of the foundations of stable homotopy theory in terms of manifolds with corners.