Samsung has been working hard at giving people less and less reason to root its recent devices, as well as making it harder to do so, but for those who want to root their Exynos-based Galaxy Note 8 units, there is now a method. Thanks to a file created by XDA Developers veteran user and recognized developer dr. ketan, rooting your Galaxy Note 8 is as simple as setting up ODIN and flashing a custom recovery with it, then using that to flash SuperSU. The catch is that your device has to have the "OEM Unlock" option checked in Developer Settings, an option that may or may not be available depending on a number of factors such as carrier and software version.
If you are able to check that option on your Exynos-based Galaxy Note 8, you can turn your phone off now and reboot into Download Mode by holding the Power, Volume Down, and Bixby buttons. It is recommended to ensure any content you want to keep is backed up beforehand. If you have ODIN all set up, download the TWRP tar image for the Galaxy Note 8 from the XDA thread in the source link, then use ODIN to flash that. If you don't already have ODIN set up, download it, then hook your phone up to your PC while the phone is on in order to install the necessary USB drivers. Once TWRP is installed, grab a copy of the SuperSU download found in the same thread, and stick it on your external SD card, if you haven't already, and flash it. Once your device boots up, you'll have fully functional root privileges and an app to manage them.
There are a few caveats to this method at present. For starters, it can erase all of your user data. Second, the EFS partition, which contains instructions for how the phone handles essential functions, should be backed up before this procedure, and at present, dr. ketan is working on an EFS backup tool. If you don't have another way of backing it up and don't want to risk bricking your phone, you would do well to wait for that tool to be completed before using this method to root your device. Finally, it should go without saying at this point that rooting the device will cause all kinds of headaches in exchange for the benefits; no Pokemon GO, no Android Pay, and added security risks are just the tip of the iceberg. For the time being, there is no such news for those with Snapdragon-based Galaxy Note 8 units, and no way to flash custom ROMs on any variant.