In this investigation, the bandgaps and nonreciprocal transmission of the nonlinear piezoelectric phononic crystal and elastic wave metamaterial are studied. Analytical solutions for the wave motion equations with the electro-mechanical coupling are obtained. According to the continuous conditions, the stop bands and transmission coefficients of both fundamental wave and second harmonic are derived by the stiffness matrix method. Some particular examples are presented to show the nonreciprocal transmission of the nonlinear elastic waves. Additionally, nonlinear ultrasonic experiments are applied to verify the theoretical analyses and numerical simulations. This work is intended to be helpful in the design and fabrication of devices of the elastic wave diode with piezoelectric materials.