  Noncommutative tori are among historically the oldest and by now the most
developed examples of noncommutative spaces. Noncommutative Yang-Mills theory
can be obtained from string theory. This connection led to a
cross-fertilization of research in physics and mathematics on Yang-Mills theory
on noncommutative tori. One important result stemming from that work is the
link between T-duality in string theory and Morita equivalence of associative
algebras. In this article we give an overview of the basic results in
differential geometry of noncommutative tori. Yang-Mills theory on
noncommutative tori, the duality induced by Morita equivalence and its link
with the T-duality are discussed. Noncommutative Nahm transform for instantons
is introduced.
