  A mathematical introduction to the classical solutions of noncommutative
field theory is presented, with emphasis on how they may be understood as
states of D-branes in Type II superstring theory. Both scalar field theory and
gauge theory on Moyal spaces are extensively studied. Instantons in Yang-Mills
theory on the two-dimensional noncommutative torus and the fuzzy sphere are
also constructed. In some instances the connection to D-brane physics is
provided by a mapping of noncommutative solitons into K-homology.
