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Gecko is the name of the layout engine developed by the Mozilla Project. It was originally named NGLayout.
Gecko's function is to read web content, such as HTML, CSS, XUL, JavaScript, and render it on user's screen or print it. In XUL-based applications Gecko is used to render the application's user interface as well.
Gecko is used in many applications, including a few browsers, such as Firefox, SeaMonkey, etc. (for a complete list, please refer to Wikipedia's article on Gecko.) Products using the same version of Gecko have identical support for standards.