  We study the generalized supersymmetric $t-J$ model with boundaries in three
different gradings: FFB, BFF and FBF. Starting from the trigonometric R-matrix,
and in the framework of the graded quantum inverse scattering method (QISM), we
solve the eigenvalue problems for the supersymmetric $t-J$ model. A detailed
calculations are presented to obtain the eigenvalues and Bethe ansatz equations
of the supersymmetric $t-J$ model with boundaries in three different
backgrounds.
