Inspired by a recent posting of Lazaroiu: hep-th/0507222 "On the non-commutative geometry of topological D-branes", the talk will try to convey the basic ideas, not the technical details. $A_\infty$-categories were introduced by Fukaya in relation to Morse theory as a topological field theory, although the pre-existing example was the space of path in a topological space. After the examples and the formal definitions, application to D-brane string theory will be the main focus.