The field of nanoparticle based therapeutics is still in infancy. That being said, most of the times nanoparticles are used as carriers of drugs that are already out there. Therefore the FDA process of evaluating such types of multiplexed system has been constantly evolving as compared to the usual and quite lengthy preclinical and clinical trials that are typically carried out for new drugs. Again, there is no simple walk through from A through Z for regulatory approval of each nanoparticle. You may search online, and look for your particular questions at the FDA website.
Europe (EU): When they have (or are intended to) a pharmacological effect the fall under the usual regulations for pharmaceuticals. If they are used as e.g. cosmetics they do not.
For the European regulatory perspective please see: