The relatively recent phenomenon of high-frequency trading has had a profound impact on the micro-structure of financial markets. Several authors hailed it as a provider of liquidity and a mechanism for controlling volatility, two highly welcome features, especially beneficial to retail traders, whereas other authors view the situation generated by algorithmic trading as damaging for both small and institutional traders, and the orderly functioning of the markets. This paper analyzes the impact of high-frequency trading in respect of the main parameters affecting market quality: volatility, transaction costs, liquidity, price discovery, penalization of slower traders, and impact on sudden financial crises, the notorious flash crashes. As often happens within the financial community, different views stand to each other and no conclusive agreement on the value of most parameters has been reached as yet. A section on the apparently falling profits of high-frequency traders, as denounced in recent times, completes the review.