Background
Shakespeare's history plays depict events leading to and during the War of the Roses (1455-1485), a protracted struggle for power fought among the English nobility. The war takes its name from the emblems of two rival houses, the red rose (Lancaster) and the white rose (York). Both houses were descended from a common royal ancestor, Edward III.
Edward III (1312-1377) came to the throne when he was 14. His grandfather, Edward I (reigned 1272-1307) was the "Edward Longshanks" whose attempts to subjugate Scotland are depicted in the movie Braveheart. The film portrays his son Edward II (reigned 1307-27) in homophobic terms, as a weakling unworthy of rule, and suggests that his son was begotten by William Wallace (who in fact died before Edward II was even married). The political boldness of Edward II's wife Isabella is, however, captured by the film. With her lover Roger Mortimer, she deposed and murdered her husband and put her young son on the throne as a puppet. Edward III, however, staged a coup of his own, imprisoning his mother, executing her lover, and taking power into his own hands. These events form the basis of Christopher Marlowe's play Edward II (c. 1593), which was deeply influential on Shakespeare and a masterpiece in its own right.
During the fifty year reign of Edward III, England made important conquests in France. (The English desire to regain these French lands is the central motive for the action of Shakespeare's Henry V.) Edward III had five sons who lived to maturity, the eldest of whom, Edward the Black Prince (so-called from his armour) was the victor over the French at the Battle of Crécy (1344). It is prudent for a king to have two sons, one to inherit the throne and a second one in case the first one dies (witness the current young princes William and Edward, known as "the heir and spare"). Indeed, Henry VIII came to the throne in 1509 only because his older brother Arthur had died. Likewise, Charles I succeeded his father James I in 1625 because of the death of his older brother Henry. But five sons is at least three too many, since the succession of one will leave the others with nothing good to do.
To make matters worse, the Black Prince died before his father, so that the throne fell to the Black Prince's ten-year-old son Richard II in 1377. Richard's powerful uncles became the leaders of factions in a struggle for influence over the young king and power over the country. The most important among them were John of Gaunt, duke of Lancaster, and Edmund, duke of York. The battles over the throne between the descendents of these two dukes provide the material for Shakespeare's history plays.
Those plays are divided into two groups of four plays each, known as the First Tetralogy, written between 1588 and 1593, and the Second Tetralogy, written from 1597 to 1599. Ironically, the First Tetralogy deals with later events, from 1422 to 1485, during the reigns of Henry VI, Edward IV, and Richard III, and culminating in the establishment of the Tudor dynasty by Henry VII. The Second Tetralogy forms a sort of prequel, dealing with the deposition of Richard II in 1400 and the reigns of first two Lancastrian kings, Henry IV and Henry V.
Who's Who in Shakespeare'e Richard III
Richard III depicts the period of ascendency of the House of York. It opens with Richard, Duke of Gloucester, celebrating his brother's conquest of the throne, and ends with Richard's own fall at the hands of Henry, Earl of Richmond, the leader of the Lancastrians.
House of Lancaster

Both King Henry VI and Prince Edward are murdered before Richard III begins. The murder of Prince Edward is depicted in 3 Henry VI V. v, and that of Henry VI in 3 Henry VI V. vi. The wooing of Anne Neville by Richard of Gloucester in Richard III I. ii. Takes places over the corpse of Henry VI in Shakespeares text, but Olivier and McKellen substitute the corpse of Edward in their film versions. After the extinction of the direct Lancastrian line, the aspirations of the party focused on Henry, Earl of Richmond, whose father, Owen Tudor, was the child of Henry V's widow, Katherine of France, and whose mother, Margaret Beaufort, was a descendent of John of Gaunt, duke of Lancaster. Their marriage and Henry of Richmond's naming were calculated to produce a candidate for the throne.
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House of York

In Shakespeare's play, Richard III describes the murders of his father, Richard Duke of York, and his brother Rutland at the hands of the Lancastrian queen, Margaret of Anjou. In his search for political legitimacy, Richard first marries the widow of the Lancastrian heir, Edward Prince of Wales, and then tries to marry the daughter of his brother Edward IV. From the sixteenth century until recently, the murders of Edward V and his brother have been laid at the feet of Richard III, though some historians have recently questioned this and suggested that Henry VII had as much to gain from their deaths.
Henry VII's defeat of Richard III at Bosworth in 1485 gave him military control of England. His subsequent marriage to Elizabeth of York established the Tudor dynasty, and their descendents ruled by right of conquest from their father, and by right of descent from their mother. Their emblem, the Tudor rose, combined characteristics of the Lancastrian and Yorkist badges.
Henry VII was succeeded by his son Henry VIII in 1509, and Henry VIII's three children, Edward VI, Mary, and Elizabeth, each ruled. Shakespeare was born in the seventh year of Elizabeth's reign, and wrote both tetralogies during the period when she was at the zenith of her power.