Felonies that would disclude you from owning a weapon, or a history of violence, like domestic abuse.
Are you worried about something?

As long as you haven't been covicted of a violent crime or on probation getting a rifle or shotgun probably won't be a problem. Handguns are a bit different, and I think are a bit more strict. Where I live you actually have to get a paper signed by the local police department saying they are ok with you owning one.
However in ohio all they do is swipe your DL to see if it comes up clear. Used to be a 7 day waiting period, I don't know when that changed.
Paul