  This is an expository article on the A-side of Kontsevich's Homological
Mirror Symmetry conjecture. We give first a self-contained study of
$A_\infty$-categories and their homological algebra, and later restrict to
Fukaya categories, with particular emphasis on the basics of the underlying
Floer theory, and the geometric features therein. Finally, we place the theory
in the context of mirror symmetry, building towards its main predictions.
