Why is this GC system bad?

I have thought of a GC which I can't see major flaws with, and I am wondering why it isn't used more prevalently, or why I haven't heard of its use.

The system is:

All objects have a 4 byte unsigned int counter attached to them. (Could be 2 bytes, I don't know.
Whenever an object is constructed, its counter is started at 1.
When an object is sent to a function as a parameter, its counter is incremented.
When an object reaches the point where it is no longer used within a function (Could be the very end of scope) its counter is decremented.
When the object's counter reaches zero, it is deleted, as at no position in code is it referenced.

Is there a fringe case in which the counter becomes faulty? What are the disadvantages, and the advantages?

Thank you in advance for your assistance.

Solution

Its called reference counting. I would suggest reading the wikipedia article as it covers the advantages and disadvantages.