This paper studies correlations between the strategies of high-frequency trading (HFT) firms, which is a manifestation of the extent of competition in which these firms engage when pursuing similar strategies. We use a principal component analysis to show that there are several underlying common strategies and that the competing HFT firms pursuing these strategies generate most HFT activity. We investigate whether competition between HFT firms creates a systematic return factor, but find no supporting evidence for such an influence. However, the short-interval return volatility of most stocks loads negatively on a market-wide measure of correlated HFT strategies. The mitigating impact of HFT competition on stock volatility appears to be driven at least in part by a market-making strategy. We further document a negative relationship between two forms of competition—that between HFT firms and that between trading venues. We investigate a potential driver behind this negative relationship, and show that greater HFT competition within a trading venue helps smaller trading venues become more competitive or viable in terms of posting better prices and narrower spreads.