In the UK, this is known as the MOT test (Ministry of Transport), in France it's known as Contrôle technique and in Germany it's known as TÜV (Technische Überwachungsverein) after the name of an organization that does it on behalf of the state governments. In the US, the state Departments of Motor Vehicles have mandatory periodic inspections for road vehicles.
A new vehicle must be inspected after four years of registration at the latest. After that, every vehicle must be inspected every other year until the age of 10 years, after which it must be inspected every year.
I've heard the requirements are more stringent in Germany than in most European countries.
The inspections used to be done by the government. Since the 1994, they've been outsourced to private companies. They're supervised closely. In my experience, the biggest difference between them is the price and the availability of appointments at convenient times. They all offer the same coffee and donuts for the customers. The entire notion of free coffee and donuts at the annual car inspection would've been absurd before 1994. :D
This is the mechanical inspection procedure used in Finland. This video put out by the Finnish Transport and Communications Agency doesn't delve into the emissions test requirements.
