Out of the box, the stock debian distro for my Beaglebone Black didn't recognize my Edimax USB wifi adapter. But after plugging it into wired ethernet and doing an apt-get update ; apt-get dist-upgrade, it shows up just fine.
I got it to associate with an access point once, but I got a lot of packet loss, and haven't gotten it to work since. After an hour or two of fiddling, I'm just going to buy a different adapter.
Update: Looks like "ifconfig wlan0 up" gets it to associate with an AP. See http://credentiality2.blogspot.com/2015/09/getting-beaglebone-black-and-green-to.html