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Price impact refers to the correlation between an incoming order (to buy or to sell) and the subsequent price change. That a buy (sell) trade should push the price up (down) is intuitively obvious and is easily demonstrated empirically. Such a mechanism must, in fact, be present for private information to be incorporated into market prices. But it is also a sore reality for trading firms for which price impact induces large (but often overlooked) extra costs. Monitoring and controlling impact has therefore become one of the most active domains of research in quantitative finance since the mid-nineties. A large amount of empirical results has accumulated over the years concerning the dependence of impact on traded quantities, the time evolution of impact, the impact of metaorders, cross-impact, etc.

In this review talk, we will present some of the most salient empirical findings, and a variety of theoretical ideas that have been proposed to rationalise them. Some remaining puzzles and open problems will be discussed as well.