Does a company have to issue shares?

Yes, if it is a company limited by shares. All companies limited by shares must have at least one share issued to one shareholder at the time of incorporation. There must be at least one person/entity that owns the company.

After the company is formed you can add further shareholders or transfer the shares to another person or legal entity. There must always be at least one share held by one person or legal entity at all times.

Companies that are limited by guarantee do not have shares or shareholders. This thype of company is generally used for non profit making actuivies such as charities.