A national library is generally the official library of a country, maintained by that nation's resources. National libraries usually try to collect all publications issued in their countries. (Often they are automatically sent a copy or two of any book published/copyrighted in that country.) Hence the catalog of a national library is often the best guide to publications issued in the country in question. The Library of Congress and the British Library are well-known national libraries, and
http://www.library.uq.edu.au/natlibs/ lists many more which have web pages and online catalogs.

Is the Library of Congress really a national library?

From "Jefferson's Legacy - A Brief History of the Library of Congress" by John Y. Cole http://lcweb.loc.gov/loc/legacy/:

Since its creation, the Library of Congress has been part of the legislative branch of the American government, and even though it is recognized as the de facto national library of the United States, it does not have that official designation. Nevertheless, it performs those functions performed by national libraries elsewhere and has become a symbol of American democracy and faith in the power of learning.
(A warning about that document, it appears to be fact-based propaganda and contains very few facts that shed the Library of Congress in anything but a positive light.)

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