Quantum Simulations with Photons and Polaritons : Merging Quantum Optics with Condensed Matter Physics This book reviews progress towards quantum simulators based on photonic and hybrid light matter systems covering theoretical proposals and recent experimental work. Quantum simulators are specially designed quantum computers. Their main aim is to simulate and understand complex and inaccessible quantum many body phenomena found or predicted in condensed matter physics materials science and exotic quantum field theories. Applications will include the engineering of smart materials robust optical or electronic circuits deciphering quantum chemistry and even the design of drugs. Technological developments in the fields of interfacing light and matter especially in many body quantum optics have motivated recent proposals for quantum simulators based on strongly correlated photons and polaritons generated in hybrid light matter systems. The latter have complementary strengths to cold atom and ion based simulators and they can probe for example out of equilibrium phenomena in a natural driven dissipative setting. This book covers some of the most important works in this area reviewing the proposal for Mott transitions and Luttinger liquid physics with light to simulating interacting relativistic theories topological insulators and gauge field physics. The stage of the field now is at a point where on top of the numerous theory proposals; experiments are also reported. Connecting to the theory proposals presented in the chapters the main experimental quantum technology platforms developed from groups worldwide to realize photonic and polaritonic simulators in the laboratory are also discussed. These include coupled microwave resonator arrays in superconducting circuits semiconductor based polariton systems and integrated quantum photonic chips. This is the first book dedicated to photonic approaches to quantum simulation reviewing the fundamentals for the researcher new to the field and providing a complete reference for the graduate student starting or already undergoing PhD studies in this area. Introduction; Dimitris G. Angelakis 1 Strongly correlated polaritons in nonlinear cavity arrays; Andrea Tomadin Davide Rossini Rosario Fazio 2 Phase diagram and excitations of the Jaynes Cummings Hubbard model; Sebastian Schmidt and Gianni Blatter 3 Out of equilibrium physics in driven dissipative photonic resonator arrays; Changsuk Noh Stephen R. Clark Dieter Jaksch Dimitris G. Angelakis 4 Topological physics with photons; Mohammad Hafezi and Jacob Taylor 5 Exciton Polariton Quantum Simulators; Na Young Kim and Yoshihisa Yamamoto 6 Strongly correlated photons in quantum photonic platforms; D. Gerace C. Ciuti and I. Carusotto 7 Quantum simulations with circuit quantum electrodynamics; Guillermo Romero Enrique Solano and Lucas Lamata 8 Dirac Dynamics in Waveguide Arrays: From Zitterbewegung to Photonic Topological Insulators; F. Dreisow M. C. Rechtsman J. M. Zeuner Y. Plotnik R. Keil S. Nolte M. Segev and A. Szameit Glossary.