Abstract: I will explain the construction of the Fukaya-Morse category of a Riemannian manifold X—an A-infinity category (a category where associativity of composition holds only "up-to-homotopy") where the higher composition maps are given in terms of numbers of embedded trees in X, with edges following the gradient trajectories of certain Morse functions. I will give simple examples and explain different approaches to understanding the structure and proving the quadratic relations on the structure maps—(1a) via homotopy transfer, (1b) effective field theory approach, (2) topological quantum mechanics approach. The talk is based on a joint work with O. Chekeres, A. Losev, and D. Youmans, arXiv:2112.12756.