Is it legal to be a real-life superhero?

Take Batman, for instance. He doesn't have any superpowers (except for money). If someone wanted to be like him, trained, and bought lots of gadgets, and started subduing criminals on the streets, would it work? Would the police look for him? If he didn't kill the criminals, and if he had a license for all of the dangerous gadgets, would he be arrested still? I've heard of something like a citizen's arrest. Would that factor affect anything?