A Statutory Corporation is a corporation created by statute. Their precise nature varies by jurisdiction thus they might be ordinary companies/corporations owned by a government with or without other shareholders, or they might be a body without shareholders which is controlled by national or sub-national government to the (in some cases minimal) extent provided for in the creating legislation.
Bodies described in the English language as "statutory corporations" exist in the following countries in accordance with the associated descriptions (where provided) :-

Australia

Canada
An example of a Statutory Corporation is a "Kassenärztliche Vereinigung", a body involved in the provision of out-patient medical services in a German state.

India
Main article: State-sponsored bodies of the Republic of Ireland.
In the Republic of Ireland, a statutory corporation is a body corporate, which is created under a particular Act of the Oireachtas, which is expected to operate as if it were a commercial company (with or without a subsidy from the Exchequer, depending on whether or not it would make a profit without one). Such bodies do not have shareholders, but are typically boards appointed by a sponsor minister. The provisions of the Companies' Acts do not typically directly apply to such bodies, although their founding legislation may specify similar requirements. As they are not formally companies they do not make a profit or loss, but rather a surplus or deficit.
The statutory corporation format was usually the form most state-sponsored bodies of the Republic of Ireland took until recent years, however usual policy today is that a private limited company by shares or public limited company incorporated under the Companies' Acts is set up instead, with the relevant Minister holding 100% of the issued share capital. Nonetheless as of 2007 several promient statutory corporations continue to exist, such as Radio Telefís Éireann, the Electricity Supply Board, and Bord Gáis Éireann

United Kingdom
At the state level, municipal corporations and counties are often created by legislative acts. Some organizations such as a transit districts or special purpose corporations such as a university, are also created by statute. In some states, a city or country can be created by petition of a certain number or percentage of voters or landholders of the affected area, which then causes a municipal corporation to be chartered as a result of compliance with the appropriate law. Corporations to be established for most other purposes are usually just incorporated as any other non-profit corporation, by filing the paperwork with the appropriate agency as part of the formation of the entity.
At the Federal level, a small number of corporations are created by Congress. Prior to the District of Columbia being granted the ability to issue corporate charters in the late 1800s, corporations operating in the District required a congressional charter. With limited exceptions, most corporations created by Congress are not federally chartered, but are simply created as District of Columbia corporations as a result of the enabling law.
There are a number of federally chartered corporations that still exist, some relatively famous ones include the Boy Scouts of America, each of the Federal Reserve Banks, and the Federal Deposit Insurance Corporation. The basic advantage for being federally chartered is that no other corporation anywhere in the United States is allowed to have the same name.