This research has worked to develop a general computer implementation for symplectic algorithms applied to the N-body problem. A program was written to evolve an arbitrary N-body problem with an arbitrary symplectic integration algorithm so long as the coefficient matrix corresponding to the algorithm is known. After the implementation was completed, a set of sample two and three body systems were evolved using the symplectic integration methods. It was found that high accelerations cause the symplectic algorithms to fail in terms of accuracy. It was also found that while high order integrators struggled to accurately evolve a system with very long time steps, they far surpassed low order methods as the time step was reduced. It was also found that the symplectic integrators were better able to evolve N-body problems over long time intervals than the backward differentiation formula methods.