We construct and solve the boundary Yang-Baxter equation in the RSOS/SOS representation. We find two classes of trigonometric solutions; diagonal and nondiagonal. As a lattice model, these two classes of solutions correspond to RSOS/SOS models with fixed and free boundary spins, respectively. Applied to (1+1 )-dimensional quantum field theory, these solutions give the boundary scattering amplitudes of the particles. For the diagonal solution, we propose an algebraic Bethe ansatz method to diagonalize the SOS-type transfer matrix with boundary and obtain the Bethe ansatz equations.