I like dogs, cats, and birds.
or
I like dogs, cats and birds.

Which is correct? Both are. Whether you use the serial comma is entirely up to you. 

The Chicago Manual of Style advises that you always use a serial comma. The AP Stylebook advises against the serial comma, unless it's required to clarify the meaning of the sentence. The key is to be consistent. Make a decision on which style you'll follow and stick to it throughout your document. Inconsistency is the grammar mistake.

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