  In this paper we discuss D-branes in the Melvin background and its
supersymmetric generalizations. In particular we determine the D-brane spectra
in these backgrounds by constructing their boundary states explicitly, where
some of the D-branes are supersymmetric. The results sensitively depend on
whether the value of magnetic flux in the Melvin background is rational or
irrational. For the rational case the D-branes are regarded as the
generalizations of fractional D-branes in abelian orbifolds ${\bf C^n/Z_N}$ of
type II or type 0 string theory. For the irrational case we found a very
limited spectrum. Since the background includes the nontrivial H-flux, the
D-branes will provide interesting examples from the viewpoint of the
noncommutative geometry.
