Succession to the many thrones often did not pass smoothly from parent to child; lack of heirs, civil wars, murders and invasions affected the inheritance in ways that a simple list does not show. The relationships that formed the basis for claims to throne are noted where we know them, and the dates of reign indicated.
The House of Stuart
Engaged to the Dauphin at age five, Mary Stuart, Queen of Scots was thus brought up in the French court where she became "Marie Stuart, Reine de l'Écosse," etc., to render the sound of 'Stewart' into French as accurately as possible. Mary kept the French spelling on her return to Scotland in 1560.
Monarchs of England, Scotland, and Ireland
In 1603, James VI of Scotland inherited the English throne upon the death of Elizabeth I in what is known as the Union of the Crowns. From then until 1707, England, Scotland, and Ireland had shared monarchs.
The Period of Interregnum, (Commonwealth and Protectorate)
England had no king from 1649 to 1660, but the constitutional status of the government was never clear. For example, the Long Parliament, up until its dissolution on 20 April 1653, was commonly recognised as a Republic. It, however, styled itself as a Commonwealth. Following the dissolution of the Rump, a Nominated Assembly was formed. Not until Cromwell accepted the Instrument of Government on 15 December 1653 did the constitutional status of the regime change. From then on Oliver Cromwell was styled as Lord Protector, ruling through two Protectorate Parliaments. In 1659, Richard Cromwell abdicated, returning power to Parliament until the Stuart Restoration in 1660.
The House of Hanover
Under the Act of Settlement 1701, the English (thus, the successor British) throne could only be held by a Protestant. Sophia of Hanover, the nearest such relative, thus became statutorily designated as the next heir. She died shortly before Anne, and her place was taken by her son, who thus founded the House of Hanover (aka Guelph and Brunswick).
Monarchs of the United Kingdom of Great Britain and Northern Ireland
In 1922, the Irish Free State left the United Kingdom. The name of the Kingdom was amended in 1927 to reflect the change. Between 1927 and the passage of the Republic of Ireland Act in 1949 George V, Edward VIII, and George VI were also styled "King of Ireland".