The licensing and registration requirements - if any - that affect your business will depend on the nature of the business and the activities it undertakes. For example, authorisation is required to carry out certain business activities involving consumer credit. And certain types of business need to register for anti-money laundering purposes with their regulating authority (which varies according to industry sector).
In many cases a professional firm (or at least the individuals working within it) will need to obtain authorisation from a professional or regulatory body.
Trade associations and professional bodies are a good starting place for finding out more information about licensing requirements. The Trade Association Forum website contains a directory of trade and professional associations.
Businesses that keep computerised records which could enable individuals - either staff or customers - to be identified may need to register as data users with the Information Commissioner's Office (ICO). There is a small fee for registering. From May 2018 the General Data Protection Regulation has introduced additional protection for personal data. You can find out more on the ICO website.
Selling general insurance
Anyone who sells, advises on, arranges or assists in selling general insurance may need to be regulated by the Financial Conduct Authority (FCA) even if insurance is only a small part of their business. If this applies to your business you can either become directly authorised by the FCA or act as an 'appointed representative' of an authorised principal. Contact the FCA for further information.