Recently a new type of system exhibiting spontaneous coherence has emerged-the exciton-polariton condensate. Excitonpolaritons (or polaritons for short) are bosonic quasiparticles that exist inside semiconductor microcavities, consisting of a superposition of an exciton and a cavity photon. Above a threshold density the polaritons macroscopically occupy the same quantum state, forming a condensate. The polaritons have a lifetime that is typically comparable to or shorter than thermalization times, giving them an inherently non-equilibrium nature. Nevertheless, they exhibit many of the features that would be expected of equilibrium Bose-Einstein condensates (BECs). The non-equilibrium nature of the system raises fundamental questions as to what it means for a system to be a BEC, and introduces new physics beyond that seen in other macroscopically coherent systems. In this review we focus on several physical phenomena exhibited by exciton-polariton condensates. In particular, we examine topics such as the di erence between a polariton BEC, a polariton laser and a photon laser, as well as physical phenomena such as superfluidity, vortex formation, and Berezinskii-Kosterlitz-Thouless and BardeenCooper-Schrie er physics. We also discuss the physics and applications of engineered polariton structures.S pontaneous coherence is a phenomenon that has fascinated physicists from a wide range of fields, ranging from condensed matter physics, atomic physics and quantum optics, to highenergy physics. Lasing is perhaps the most ubiquitous phenomenon giving rise to macroscopic coherence, in this case formed by stimulated emission of photons 1 . Bose-Einstein condensation (BEC) is another example of collective coherence of many particles, such that above a critical density (or equivalently below a critical temperature), the particles spontaneously occupy the ground state 2 . Superfluid 4 He is the earliest realization of BEC, in the presence of strong interactions 3 . Superconductivity, viewed as a condensation of Cooper pairs, allows a charged version of BEC yielding resistanceless (superfluid) flow 4 . The aspect that is common to these phenomena is that a large number of particles initially possessing no phase relation all become coherent once a system parameter, such as temperature or density, crosses a threshold. In this review, we examine a new system that undergoes spontaneous coherence: the exciton-polariton condensate. The recent observation of excitonpolariton condensation 5-7 adds another particle to the list for which BEC has been observed-cold atoms 8,9 , magnons 10,11 , and more recently photons 12 . We shall see that rather than being simply another type of particle that undergoes BEC, it possesses characteristics that incorporate new physics owing to its intrinsically non-equilibrium nature.One of the distinctive features of exciton-polaritons (or simply polaritons for short) is their exceedingly light effective mass, typically of the order of 10 −4 times the bare electron mass. For an ideal (non-interacting and a...