I discuss recent developments of the study of “synthetic dimensions” in photonics. The idea of synthetic dimensions is to identify internal states of a photonic cavity as extra dimensions, and to simulate higher dimensional lattice models using physically lower dimensional systems. The concept was originally proposed and experimentally realized in ultracold gases [1–5]. I first review the existing theoretical and experimental studies of synthetic dimensions. After discussing some challenges and limitations of the existing methods of synthetic dimensions, I explain our proposals of realizing synthetic dimensions in photonic cavities [6, 7], which overcome some of these limitations. Finally I discuss how the four dimensional quantum Hall effect can be observed in photonics using the synthetic dimensions [6, 8, 9]. [1] O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, Quantum Simulation of an Extra Dimension, Phys. Rev. Lett. 108, 133001 (2012). [2] A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzelinas, and M. Lewenstein, Synthetic Gauge Fields in Synthetic Dimensions, Phys. Rev. Lett. 112, 043001 (2014). [3] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, Observation of chiral edge states with neutral fermions in synthetic Hall ribbons, Science 349, 1510 (2015). [4] B. K. Stuhl, H. I. Lu, L. M. Aycock, D. Genkina, and I. B. Spielman, Visualizing edge states with an atomic Bose gas in the quantum Hall regime, Science 349, 1514 (2015). [5] L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati, M. Frittelli, F. Levi, D. Calonico, J. Catani, M. Inguscio, and L. Fallani, Synthetic dimensions and spin-orbit coupling with an optical clock transition, Phys. Rev. Lett. 117, 220401 (2016). [6] T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics Phys. Rev. A 93, 043827 (2016). [7] T. Ozawa and I. Carusotto, Synthetic dimensions with magnetic fields and local interactions in pho- tonic lattices, Phys. Rev. Lett. 118, 013601 (2017). [8] H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, and N. Goldman, Four-Dimensional Quantum Hall Effect with Ultracold Atoms, Phys. Rev. Lett. 115, 195303 (2015). [9] H.M.Price,O.Zilberberg,T.Ozawa,I.Carusotto,andN.Goldman,MeasurementofChernnumbers through center-of-mass responses, Phys. Rev. B 93, 245113 (2016).