While console jailbreaking and software modding is nothing new, there are a few consoles that have long stood against hackers and exploiters. The Xbox 360 is one such console, and while a few exploits have surfaced over the years, they require either a hardware modification or the purchase and use of a specific game (which has seen a subsequent price spike on sites like eBay). That changes with the release of a new software-only hack that requires nothing more than a USB flash drive.
The exploit, dubbed FreeMyXe, runs in conjunction with another exploit known as BadUpdate. Version 1.1 of BadUpdate requires the use of either Tony Hawk's American Wasteland or Rock Band Blitz. Users can opt for either the full retail version of Rock Band Blitz or a free trial of the game, which is what FreeMyXe uses.
Essentially, the free trial of the game can be downloaded (at no cost) and transferred onto the root of a USB drive. Users can then put the exploits on the USB drive (along with any homebrew or other unsigned applications desired). There's a bit more prep work needed (detailed in the YouTube video from Modern Vintage Gamer below), but the process is fairly straightforward and (perhaps best of all) free.
The exploit requires firmware version 2.0.17559.0, and it's recommended that an exploited console should be disconnected from the Internet in order to avoid a potential console ban by Microsoft.
There is a caveat to this exploit: it is technically a tethered jailbreak, which means that if the console is powered down or crashes after the exploit has been run, it will need to be re-run after the console is powered back on. The exploit also takes roughly 15-20 minutes to complete. However, it is probable that this exploit may be optimized and could result in an untethered jailbreak, resulting in a true softmod.