One of the worst things they are doing is "quote stuffing", which means submitting bids and cancelling them tens of thousands of times per second. They stuff the exchanges with quotes and eventually cause a huge latency with respect to the quotes that are shown by the exchange, such as NYSE, NASDAQ, etc. However, they aren't affected because they can see the prices because they have a raw data feed. This gives them an edge relative to the other traders that don't have their own raw feed straight from the exchanges, and they use that to basically shoot fish in a barrel.
This was one of the main causes for the flash crash in 2010. The excessive quotes
The thing is that they could stop this by implementing a minimum time for a quote to exist, something innocuous like 100 ms, but they don't because this would be a detriment to the HFT shops.
The other thing is that HFT shops behave like market makers, and they claim to provide liquidity, however, they have no obligation to provide liquidity, like real market makers. So just like during the flash crash, they turned off their machines, and the markets went haywire. So they want their cake and eat it too. If they are going to trade like market makers, they need to have the obligation to continue to provide liquidity.