The Fukaya category of open symplectic manifolds is expected to have good local-to-global properties. Based on this idea several people have developed sheaf-theoretic models for the Fukaya category of punctured Riemann surfaces: the name topological Fukaya category appearing in the title refers to the (equivalent) constructions due to Dyckerhoff-Kapranov, Nadler and Sibilla-Treumann-Zaslow. In this talk I will introduce the topological Fukaya category and explain applications to Homological Mirror Symmetry for toric Calabi-Yau threefolds. This is joint work with James Pascaleff.