As the title states, I think I have a grounding problem. Whenever I plug my bass into an amp and don't touch any metal part, it buzzes. When I touch the strings, bridge, tuners, or any metal part of the bass, the buzzing stops. The wiring looked just fine when I looked at it. Are there any other possibilities?
Oh crap!
Have you taken the bass apart? Yeah, it's a definitly a grounding problem somewhere. There should be a wire in your bass thats not connected to anything just soldered to a piece of metal. Sometimes it's underneath a back plate if you have one. It probably just came undone or something. But yeah, its just a wire that gets soldered to a random piece of metal I believe. I'm quite positive that that's your problem.

Yeah, I think that's my problem too. I even took the bridge off and soldered the wire there to the bridge. It didn't totally stick, but it's touching the bridge, so it's grounded there. But the buzz still occurs, and I don't know why.
Oh crap!
If it's just touching the bridge, it might not be enough. If it's not soldered on, you need to make sure that there's enough contact force between the ground wire and the bridge. Otherwise, as long as everything is wired up correctly, it could be something else (though I'm not sure what that something else would be).
Check the output jack. If you only have two posts, make certain you've soldered to the right ones. If you have 3 posts (but a mono jack), try switching the GND to the other post. I had this same problem with my bro's guitar, i had to use a random jack lying around. It had 3 posts, one was hot (tip), one was linked to the outside of the plug, and there was a ring on the inside. However, it was only a mono jack. I assumed that the inner ring and the outside post were the same (i thought they were contacting eachother), but apparently they were'nt. Try switching those out.
sounds like a normal problem to me unless you shielded your bass...if your around a lot of electronics it could also do that