A bumper apparatus with a multistage energy absorbing structure mounted to the front and rear ends of a vehicle. The bumper apparatus includes a back beam, a rear shock absorber, a front shock absorber, and a bumper cover. The back beam is mounted to a car body. The rear shock absorber is mounted to the front surface of the back beam and has a plurality of rear shock absorbing units on a middle portion thereof. The front shock absorber is mounted to the front surface of the rear shock absorber and has an arcuate cross-sectional structure. The bumper cover is mounted to the front surface of the front shock absorber.