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 transport(1-5). 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 moments(6-8). 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.