By the Grace of God
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For example, according to the "Royal Proclamation reciting the altered Style and Titles of the Crown" of May 29, 1953, Elizabeth II's full title is "Elizabeth II, by the Grace of God, of the United Kingdom of Great Britain and Northern Ireland and of Her other Realms and Territories Queen, Head of the Commonwealth, Defender of the Faith"; in the various other Commonwealth Realms, variations are used, specifying the realm in question and varying some of the other elements of the title.
History and rationale 
Originally, it had a literal meaning: the divine will was invoked—notably by Christian monarchs—as legitimation (the only one above every earthly power) for the absolutist authority the monarch wielded. This is also known as the divine right of kings, that is, the endorsement of God for the monarch's reign.
While the Christian Roman emperors during the late Dominate, especially in the East (as continued in Byzantium after the fall of Rome), came remarkably close to acting out the role of God's voice on earth, centralizing all power in their hands, e.g. reducing the Patriarch of Constantinople to their "(State) Minister of the Cult" and proclaiming their "universal" authority (in the Oriental tradition, as in Persia, but also in the original Muslim Caliphate), for most dynasties it would rather prove to be a never-ending battle up the hills of political resistance, both from rival power poles within their state (nobility, clergy, people; even within a dynasty) and from foreign powers claiming independence or even hegemony, usually constraining them in constitutional limitations (not necessarily written statutes, more often a matter of customary law and established privileges).
By custom, the phrase "by the Grace of God" is restricted to sovereign rulers; in the feudal logic, a vassal could not use it, because he held his fief not by the grace of God almighty, but by grant of a superior noble, (in)directly from the crown. Yet this did not stop kings to continue using it, even when some of them did homage to the pope (as viceregent of God) and/or (an)other ruler(s) (sometimes even mutually), on account of some (minor or 'external') fief, or even for their actual principality, such as the many belonging to the Holy Roman Empire.
While the "incantation" of divine Grace became a prestigious style figure that few Christian monarchies could resist, it is not a literal carte-blanche from Heaven, but rather a consecration of the "sacred" mystique of the crown. Some of that survives even in modern constitutional monarchies and finds expression in most even mildly religious republics and dictatorships, where all power has been transferred to elected (party) politicians. In modern, especially recently (re-)founded monarchies, more realistic power reports (often crucially a voice in the succession and the purse strings) do in time find expression, sometimes even in abandoning "By the Grace of God", or rather, especially earlier, in the intercalation of compensatory phrases, such as "and the will of the people", and/or replacing the genitive "sovereign of X-place" by "sovereign of the X-inhabitants", quite meaningful where linked to the Enlightenment-notion of the "social contract", which means the nominal 'sovereign' is in fact potentially subject to national approval, without which a revolution against him can be legitimate.
Today, even though all western monarchies are constitutional, all political power having passed to the people (by referendum or, generally, elections), the now hollow traditional phrase "by the grace of God" is still included in the full titles and styles of the monarchs of Denmark, Liechtenstein, the Netherlands, Spain and the United Kingdom, but not in that of Belgium, Luxembourg, Monaco, Norway and Sweden. In Luxembourg it had existed until 2000, when Henri, the current Grand Duke, decided to drop it. Like the use of the term "subject" for the citizens of a monarchy, "by the Grace of God" is a protocolary form that has survived the emancipation of the electorate from its once absolute rulers, who now rule only in name, but without real political power.
Spain's 1978 Constitution, in article 56, §2, states that the title of the King of Spain is simply "King of Spain" (Rey de España), but that he also possesses the traditional titles of the Spanish Crown (podrá utilizar los demás que correspondan a la Corona). As a result, the King of Spain continues to be King "by the grace of God". During the twentieth-century dictatorship of Francisco Franco, Spanish coins bore a legend identifying him as Francisco Franco, por la G. de Dios Caudillo de España ("by the G(race) of God, Leader of Spain").
Parallels exist in other civilizations, e.g. Mandate of Heaven of the Chinese empire, where for centuries the official decrees by the Emperors of China invariably began with the phrase (pinyin)"Fèng Tiān chéng yùn, Huángdì zhào yuē" (奉天承運 皇帝詔曰) which is translated as "By the Grace of Heaven, the Emperor decrees".
In modern languages 
- This list, possibly incomplete, is limited to phrases that are/were formally used by monarchies of the (mainly western/Christian) tradition in their official styles.
- By the Grace of God (English)
- Deur die Genade van God (Afrikaans)
- Af Guds nåde (Danish), pre-1948 spelling: Af Guds Naade)
- Bij de Gratie Gods (Dutch)
- Av Guðs/Guds náði (Faroese)
- Von Gottes Gnaden (German)
- Af Guðs náð (Icelandic)
- Av Guds nåde (Norwegian)
- Av Guds nåde (Swedish)
- Per la gràcia de Déu (Catalan)
- Par la Grâce de Dieu (French)
- Per la Grazia di Dio (Italian; in Piedmont/Sardinia and in Kingdom of Italy replaced by Per Grazia di Dio e Volontà della Nazione after the constitution of 1848)
- Por graça de Deus or Pela graça de Deus (Portuguese)
- Por la Gracia de Dios (Spanish; in Spain dropped since 1978, replaced by Rey Constitucional de España, etc.)
- Prin Harul lui Dumnezeu (Romanian) (also Din Mila lui Dumnezeu or Prin grația lui Dumnezeu)
- Milošću Božijom (Bosnian)
- Божію Поспѣшествующею Милостію (Russian archaic Cyrillic spelling)
- По Божия милост (Bulgarian)
- Z milosti Boží (Czech)
- Milošću Božjom or Božjom milošću (Croatian)
- No Dieva žēlastības (Latvian)
- Dievo malone (Lithuanian)
- По милост Божја or По Божја Милост (Macedonian)
- Z Bożej łaski (Polish)
- Po milosti Božjoj (Serbian)
- Z Božej milosti (Slovak)
- Po milosti božji (Slovenian)
- " Deu Kurpen" Konkani
- परमेश्वर कि कृपा से (Hindi)
- Parmatma Di Mehar Naal (Punjabi language)
- Devuni krupa valana (Telugu)
- Isten kegyelméből (Hungarian)
- Ἐλέῳ Θεοῦ (Eleōι Theou) (Greek)
- By the will of our Lord is an equivalent formula in Orthodox Georgia
- Jumalan armosta(Finnish)
- Trí Ghrásta Dé (Irish)
- Trwy Ras Duw (Welsh)
- Bil-Grazzja ta' Alla (Maltese)
- Lagbara Olorun (Yoruba)
Similar concepts unrelated to and sometimes predating Christianity:
- 「奉天承運」 (By the Grace of Heaven) was a style formerly used by the Emperor of China and the Emperor of Korea.
- 「天佑ヲ保有シ萬世一系ノ皇祚ヲ踐メル」 (By the Grace of Heaven, seated on the throne occupied by the same dynasty from time immemorial) was a style used by the Emperor of Japan until 1945
- بالله - Billah (through God), often attached to descriptive names of Caliphs
Compound variations on the formula 
- Louis-Napoléon Bonaparte was crowned Napoléon III, Emperor of the French By the Grace of God and National Will (Par la Grâce de Dieu, et la Volonté Nationale) after a plebiscite organized among the French people.
- Oliver Cromwell was Lord Protector by the Grace of God, and the Republic, denoting that he was chosen by God to rule but he was put there by the people of the 'Commonwealth' (British republic).
- Agustín de Iturbide of Mexico was styled Agustín I, By the Providence of God, Constitutional Emperor of Mexico
- By the Grace of God and the Will of the Nation (Per Grazia di Dio e Volontà della Nazione) in Kingdom of Italy, as well as in the Italian Empire where the king was styled By the Grace of God and the Will of the [Italian] Nation King of Italy, King of Albania, Emperor of Ethiopia which though omitted the titularity as King of Cyprus and Jerusalem which had instead styled the House of Savoy previously and alongside with Duke of Savoia, King of Sardinia, Prince of Piedmont
- By the Grace of God and the Will of People in Kingdom of Serbia and Kingdom of Yugoslavia. The same title was used in the Polish-Lithuanian Commonwealth when, starting with king Henryk Walezy and ending with king Stanisław August Poniatowski, the king was chosen by the noblemen in a free election.
- Sovereigns of the Kingdom of Hawai'i were styled "By the Grace of God and under the Constitution of the Hawaiian Islands, King (or Queen)"