The topological phases of matter are characterized using the Berry phase, a geometrical phase associated with the energy-momentum band structure. The quantization of the Berry phase and the associated wavefunction polarization manifest as remarkably robust physical observables, such as quantized Hall conductivity and disorder-insensitive photonic transport1,2,3,4,5. Recently, a novel class of topological phases, called higher-order topological phases, were proposed by generalizing the fundamental relationship between the Berry phase and quantized polarization, from dipole to multipole moments6,7,8. Here, we demonstrate photonic realization of the quantized quadrupole topological phase, using silicon photonics. In our two-dimensional second-order topological phase, we show that the quantization of the bulk quadrupole moment manifests as topologically robust zero-dimensional corner states. We contrast these topological states against topologically trivial corner states in a system without bulk quadrupole moment, where we observe no robustness. Our photonic platform could enable the development of robust on-chip classical and quantum optical devices with higher-order topological protection.