3.10 Open Edition is a custom firmware for your PSP, It is created by the world famous Dark Alex. The complete download for it is below. It was only released today so its the newest as of yet.
So what is it? It is a Combination of two official Sony firmwares, Firmware 1.5 and Firmware 3.10. So what are you getting? Well the features from 1.5 like homebrew launching ability, firmware tweaks and much more. Plus all the features from 3.10 such as web browser, flash support, rss reader, ps1 emulator, camera compatibility , ps3-psp connectivity, music visualizations, several new audio & video codecs and most important limited sony control over it.