We discuss and illustrate how symplectic integrators can be tailored to solve the time-dependent Schrodinger equation, yielding a large new family of wave packet propagation methods. These methods are interesting because of their algorithmic simplicity and minimal storage requirements. A variety of such methods are obtained. Calculations and comparisons with various other methods are presented for a one-dimensional Morse oscillator and a three-dimensional unimolecular dissociation problem.