ISBN (identifier)
In general, the ISBN check digit is calculated as follows.
Let

Then

This check system—similar to the UPC check digit formula—does not catch all errors of adjacent digit transposition. Specifically, if the difference between two adjacent digits is 5, the check digit will not catch their transposition. For instance, the above example allows this situation with the 6 followed by a 1. The correct order contributes 3 × 6 + 1 × 1 = 19 to the sum; while, if the digits are transposed (1 followed by a 6), the contribution of those two digits will be 3 × 1 + 1 × 6 = 9. However, 19 and 9 are congruent modulo 10, and so produce the same, final result: both ISBNs will have a check digit of 7. The ISBN-10 formula uses the prime modulus 11 which avoids this blind spot, but requires more than the digits 0–9 to express the check digit.
Additionally, if the sum of the 2nd, 4th, 6th, 8th, 10th, and 12th digits is tripled then added to the remaining digits (1st, 3rd, 5th, 7th, 9th, 11th, and 13th), the total will always be divisible by 10 (i.e., end in 0).

ISBN-10 to ISBN-13 conversion: A 10-digit ISBN is converted to a 13-digit ISBN by prepending "978" to the  ISBN-10 and recalculating the final checksum digit using the ISBN-13 algorithm. The reverse process can also be performed, but not for numbers commencing with a prefix other than 978, which have no 10-digit equivalent.

Errors in usage: Publishers and libraries have varied policies about the use of the ISBN check digit. Publishers sometimes fail to check the correspondence of a book title and its ISBN before publishing it; that failure causes book identification problems for libraries, booksellers, and readers. For example, ISBN 0-590-76484-5 is shared by two books—Ninja gaiden: a novel based on the best-selling game by Tecmo (1990) and Wacky laws (1997), both published by Scholastic.
Most libraries and booksellers display the book record for an invalid ISBN issued by the publisher. The Library of Congress catalogue contains books published with invalid ISBNs, which it usually tags with the phrase "Cancelled ISBN". The International Union Library Catalog (a.k.a., WorldCat OCLC—Online Computer Library Center system) often indexes by invalid ISBNs, if the book is indexed in that way by a member library.

eISBN: Only the term "ISBN" should be used; the terms "eISBN" and "e-ISBN" have historically been sources of confusion and should be avoided. If a book exists in one or more digital (e-book) formats, each of those formats must have its own ISBN. In other words, each of the three separate EPUB, Amazon Kindle, and PDF formats of a particular book will have its own specific ISBN. They should not share the ISBN of the paper version, and there is no generic "eISBN" which encompasses all the e-book formats for a title.

EAN format used in barcodes, and upgrading: The barcodes on a book's back cover (or inside a mass-market paperback book's front cover) are EAN-13; they may have a separate barcode encoding five digits called an EAN-5 for the currency and the recommended retail price. For 10-digit ISBNs, the number "978", the Bookland "country code", is prefixed to the ISBN in the barcode data, and the check digit is recalculated according to the EAN-13 formula (modulo 10, 1× and 3× weighting on alternating digits).
Partly because of an expected shortage in certain ISBN categories, the International Organization for Standardization (ISO) decided to migrate to a 13-digit ISBN (ISBN-13). The process began on 1 January 2005 and was planned to conclude on 1 January 2007. As of 2011, all the 13-digit ISBNs began with 978. As the 978 ISBN supply is exhausted, the 979 prefix was introduced. Part of the 979 prefix is reserved for use with the Musicland code for musical scores with an ISMN. The 10-digit ISMN codes differed visually as they began with an "M" letter; the bar code represents the "M" as a zero, and for checksum purposes it counted as a 3. All ISMNs are now thirteen digits commencing 979-0; 979-1 to 979-9 will be used by ISBN.
Publisher identification code numbers are unlikely to be the same in the 978 and 979 ISBNs, likewise, there is no guarantee that language area code numbers will be the same. Moreover, the 10-digit ISBN check digit generally is not the same as the 13-digit ISBN check digit. Because the GTIN-13 is part of the Global Trade Item Number (GTIN) system (that includes the GTIN-14, the GTIN-12, and the GTIN-8), the 13-digit ISBN falls within the 14-digit data field range.
Barcode format compatibility is maintained, because (aside from the group breaks) the ISBN-13 barcode format is identical to the EAN barcode format of existing 10-digit ISBNs. So, migration to an EAN-based system allows booksellers the use of a single numbering system for both books and non-book products that is compatible with existing ISBN based data, with only minimal changes to information technology systems. Hence, many booksellers (e.g., Barnes & Noble) migrated to EAN barcodes as early as March 2005. Although many American and Canadian booksellers were able to read EAN-13 barcodes before 2005, most general retailers could not read them. The upgrading of the UPC barcode system to full EAN-13, in 2005, eased migration to the ISBN in North America.

See also: ASIN (Amazon Standard Identification Number)
BICI (Book Item and Component Identifier)
Book sources search – a Wikipedia resource that allows search by ISBNs
CODEN (serial publication identifier currently used by libraries; replaced by the ISSN for new works)
DOI (Digital Object Identifier)
ESTC (English Short Title Catalogue)
ISAN (International Standard Audiovisual Number)
ISRC (International Standard Recording Code)
ISTC (International Standard Text Code)
ISWC (International Standard Musical Work Code)
ISSN (International Standard Serial Number)
ISWN (International Standard Wine Number)
LCCN (Library of Congress Control Number)
License number (East German books) (Book identification system used between 1951 and 1990 in the former GDR)
List of group-0 ISBN publisher codes
List of group-1 ISBN publisher codes
List of ISBN registration groups
SICI (Serial Item and Contribution Identifier)
VD 16 (Verzeichnis der im deutschen Sprachbereich erschienenen Drucke des 16. Jahrhunderts, "Bibliography of Books Printed in the German Speaking Countries of the Sixteenth Century")
VD 17 (Verzeichnis der im deutschen Sprachraum erschienenen Drucke des 17. Jahrhunderts, "Bibliography of Books Printed in the German Speaking Countries of the Seventeenth Century")

Explanatory notes: 

References: 

External links: 
ISO 2108:2017 – International Standard Book Number (ISBN)
International ISBN Agency – coordinates and supervises the worldwide use of the ISBN system
Numerical List of Group Identifiers – List of language/region prefixes
Free conversion tool: ISBN-10 to ISBN-13 & ISBN-13 to ISBN-10 from the ISBN agency. Also shows correct hyphenation & verifies if ISBNs are valid or not.
"Guidelines for the Implementation of 13-Digit ISBNs" (PDF). Archived from the original (PDF) on 12 September 2004.
RFC 3187 – Using International Standard Book Numbers as Uniform Resource Names (URN)
Worldwide Auto-Converter at Library of Congress