Topological photonics is an important and rapidly growing field that offers a powerful way to manipulate light. The geometric and topological ideas in this field have thoroughly changed our understanding of light propagation. Floquet topological insulators are an important branch of topological photonics, where many novel physical phenomena are waiting to be explored. This is mainly due to the condensed matter community, which utilizes periodically driven systems to reveal new phases of matter that do not exist in equilibrium systems. This paper reviews the research on Floquet topological optical insulators over the last decade, starting from a classification of their implementations as time-based modulation, space-based modulation and resonator-coupling-based modulation of Floquet topological photonic insulators. More specifically, time-modulated Floquet topological photonic insulators represent a unique class of artificial photonic structures with special topological properties, characterized by the dynamic modulation of topological phase through a periodic time function. This modulation method introduces a distinct control mechanism to photonic systems, leading to an unprecedented light transport. A key feature of time-modulated Floquet topological photonic insulators is the realization of topological edge states by breaking time-reversal symmetry. These edge states can propagate unidirectionally along the system boundaries and exhibit robustness against defects and back-scattering. Compared with traditional static topological photonic insulators, the time-based modulation method introduces a new degree of freedom in the temporal domain, thereby offering a wider range of control strategies and application potential. Given the theoretical limitations of time-modulated Floquet topological photonic insulators along with experimental constraints due to the inability of active switching frequencies to meet experimental demands, practically experimental demonstrations in this field have encountered significant restrictions. To address this challenge, researchers have initiated the exploration of alternative methodologies. Among these, space-based Floquet topological photonic insulators have emerged as a noteworthy direction. This approach employs one spatial dimension as a surrogate for the time dimension, thereby enabling the realization of pertinent topological properties in Floquet topological photonic insulators without requiring additionally high switching frequencies. Therefore, this approach has garnered significant attention and achieved substantial progress in the field. However, due to the incompatibility with integrated photonic systems, the integration of temporal or spatial modulation on a semiconductor platform remains a challenge. To address this issue, the two-dimensional structure of coupled resonator optical waveguides has emerged as a robust platform for the study of topological photonic insulators, thanks to its unique structure and compatibility with traditional photonic integrated circuits. So far, research on Floquet topological photonic insulators based on coupled resonances is garnering increasing attention. In summary, we have demonstrated the unique topological properties and amazing physical phenomena as well as the broad research progress in applications are presented from both theoretical and applied aspects. The achievements and challenges in this field are reviewed and discussed, and further research on Floquet topological photonic insulators is foreseen.