We study the nonlinear optical response of multilayer metallic nanoparticles driven by an electromagnetic wave, which can show large field enhancement, hence significantly enhancing optical processes. In addition to optical bistability (OB), we find that optical multistability (OM), which plays a more important role in some applications than OB, is achievable and can be obtained in a multilayer plasmonic nanoparticle. Our results demonstrate that owing to strong localized fields created in the core and each layer of multilayer nanoshells, which occurs in the particles at frequencies close to the surface plasmon resonance, multilayer nanoparticles are promising systems with unique optical characteristics to control the light by light at the nanometer scale. It is demonstrated that OB can be converted to OM via adjusting the wavelength of the applied field and the size of the nanoshell, and the system can manifest optical hysteresis. It is found that the optical bistable or multistable threshold and the shape of hysteresis loops are strongly dependent on the thickness of shells, the incident wavelength, the permittivity of the surrounding medium, and the composition of the core and the inner/outer layers. We also give a discussion on the impact of the exciton-plasmon interaction and the intrinsic size effect on the nonlinear optical response of multilayer spherical nanoparticles.