I will sketch the construction of the Fukaya-Morse category of a Riemannian manifold $X$ — an $A_\infty$ 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.