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What command within a notebook will return the file name of that notebook?

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You have to add a few extra commands to NotebookFileName to remove the directory.

Last@FileNameSplit@NotebookFileName[]

Hope this helps.

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    $\begingroup$ if you don't need the extension you can do it even shorter: FileBaseName@NotebookFileName[]. If you know the extension is .nb then (FileBaseName@NotebookFileName[])<>".nb" $\endgroup$ – yohbs Jul 19 '15 at 11:30
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No need to go through all the machinations of trimming/applying a cascade of functions, simply:

CurrentValue["NotebookFileName"]
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