  We analyse the moduli spaces of superconformal field theories (SCFTs). For
N=2 we find an enhanced moduli space which in geometrical terms corresponds to
tori with two independent complex structures. To explain the precise relation
with the moduli space of SCFTs on K3 surfaces as described by Aspinwall and
Morrison, we discuss some subtleties with the precise interpretation of the N=2
and N=4 moduli spaces. We also explain why in some cases the SYZ-description of
mirror symmetry as fibrewise T-duality seems to break down.
  Using gluing matrices we give an algebraic description of D-branes and
construct the corresponding boundary states. We study how isomorphisms of the
SCFTs act on D-branes. Finally we give a geometrical interpretation of our
algebraic constructions and make contact with the geometrical D-brane
categories and Kontsevich's homological mirror symmetry conjecture.
