Edward of Windsor - as he was known early in life prior to being crowned King of England.
King of England - 1327 to 1377.
King of France - claimed from 1340 to 1360
Duke of Aquitaine - 1325 to 1360
Lord of Aquitaine - 1360 to 1377.
Lord of Ireland
Knight of the Garter - Founder Sovereign of the Order of the Garter in 1348.
Earl of Chester - Created on 24 November 1312; this title was merged into the crown in 1327. 
King of England - Forced from power by his own wife Isabel and Roger Mortimer, Edward II abdicated the throne of England in favor of his son Edward of Windsor on 21 January 1327. Edward III was proclaimed King of England on 25 January 1327, and crowned at Westminster Abbey on 1 February 1327. His mother Isabella with Roger de Mortimer served as Regent of England in his place until Edward usurped the regents (and executed Roger Mortimer) on 19 October 1330.
King of France - Edward III first styled himself King of France in 1337 when Phillip VI King of France attempted to reclaim the English Duchy of Aquitaine. His claim was based on his right from his maternal grandfather Phillip IV, King of France. In 1340, he formally proclaimed himself King France, thus precipitating the 100 Years War. In 1360, a temporary peace was reached with the Treaty_of_Brétigny where Edward III renounced his claim to French throne in exchange for sovereignty over Aquitaine and other lands in France.
Duke of Aquitaine - The Duchy of Aquitaine, with fealty owed to the French king, was held by the kings of England from the time of Henry I. Reluctant to personally present himself and swear allegiance to the king of France, Edward II named his oldest son Edward of Windsor as Duke of Aquitaine on 10 September 1325.  When sovereignty was granted to Aquitaine by the Treaty_of_Brétigny in 1360, the title duke was dropped in favor of Lord of Aquitaine.
Born: 13 November 1312 at Windsor Castle, Windsor, Berkshire, England. 
Marriage and Children
Married: Phillipa of Hainault, daughter of Guillame III, Count of Hainault, on 24 January 1327/8 at York Minster in York, Yorkshire, England.  She was likely born in early (January or February) 1314.  She died on 15 August 1369 at Windsor Castle. 
Papal dispensation for the marriage granted on 30 August 1327, they being related in the 3rd degree of kindred. 
Children of Edward III and Philippa of Hainault: 
Edward of Woodstock, Princes of Wales. Commonly known as The Black Prince. Born on 15 June 1330 at Woodstock, Oxfordshire.
Mistress: Edward III had a well-known relationship with Alice de Salisbury, a lady in the household of Queen Philippa. Her ancestry is unknown but she has been shown to be a sister of a John de Salisbury. She was the wife of John de Perrers who was dead by 18 December 1362. Edward III's relationship with Alice is believed to have begun by 1364.
Children believed to be by Edward III and Alice de Salisbury: 
John de Surrey (or Southerey).
Joan de Surrey (or Southerey).
Controversy regarding children: A 2014 Y-DNA study compared the DNA of Edward III’s descendants, namely Richard III and 5 men descended from Henry Somerset, 2nd Earl of Worcester.  Richard III descended in the male line from Edward’s son Edmund of Langley, and Henry Somerset descended in the male line from Edward’s son John of Gaunt. The Y-DNA did not match. The press sensationalized the announcement to suggest that Edward III might not be the father of Edmund of Langley or John of Gaunt, or even of his other children. 
However, this is an extremely poor analysis of the data. Certainly, there was a non-paternal event(s) somewhere in the line down to his gr-gr—grandson Richard III, or somewhere in the 24 to 26 generation descent in the Somerset lines. It is not possible to tell where the non-paternal event occurred from this study. At this point, all the children of Philippa of Hainault must be presumed to be the children of Edward III, King of England.
Died: 21 June 1377 at Sheen Palace (now Richmond), co. Surrey, England. 
From Royal Tombs of Medieval England: "Edward III died at Sheen Palace on 21 June 1377. The king's will dated the previous year instructed his burial at Westminster in the manner befitting a king. Edward's will endowed a chantry for himself, and his wife, Philippa of Hainault, at his father's foundation at King's Langley, but not at Westminster. His eldest surviving son, John of Gaunt, the keeper of the privy seal, was one of his executors. Edward was buried at Westminster only two weeks after his death, on July 5, dressed in formal robes and regalia. Edward's tomb stands immediately south of the Confessor's shrine in the bay adjacent to the monument for his wife, Philippa of Hainault, and has a gilt-bronze effigy and Purbeck marble tomb-chest with gilt-bronze dynastic figures, and heraldic wall paneling. Contemporary records claimed the king's body was interred beside Philippa, but the locations of the royal coffins has never been identified. The position of Edward's tomb was almost certainly reserved by the king himself, given his instructions for the arrangement of Westminster burials made as early as 1339. Work on the monument, though, appears to have been delayed for several years. In 1386 a ship was carrying Purbeck marble 'for the king's tomb' (with its sailors exempted from impressment) suggesting that tomb work was then only at a preliminary stage. The tomb-chest at least must have been completed by 1395 when it was used as a model for the tomb of Richard II and Anne of Bohemia." 
In 1333 he reversed Isabel's and Mortimer's policy of peace with Scotland by invading it, reviving the ambitions of his grandfather, King Edward I. Edward III's main foreign preoccupation, however, from 1337 onwards was France, whose king, Philippe VI, then declared his Duchy of Gascony forfeited. Edward formally assumed the title of King of France in right of his mother in Jan. 1340. In June 1340 the English fleet defeated the French navy in the Battle of Sluys, off the coast of Flanders. This victory gave the English control of the English Channel for the next generation. Near continuous war ensued with some respite from truces. The army, commanded by King Edward III and his son, Edward, defeated a larger French force at the battle of Crecy in August 1346, the victory owning to superior tactics and to the invention of the longbow, which decimated the mounted French knights. The financial burden of the war roused resentment, which was assuaged somewhat when Edward negotiated the main war taxes with the representatives of the shires and the borough communities sitting in parliament. He aroused enthusiasm for the war by engaging the chivalrous interests of the nobles in it and stirring up distrust and hatred of the French.
His son, Edward, won a great victory at Poitiers in Sept. 1356, capturing the French king, Jean II. In 1360 King Edward concluded the treaty of Bretigny, giving up his claim to the throne of France, receiving in turn the duchy of Aquitaine, together with Calais, Guines, Poitou, and Ponthieu in full sovereignty.
In the war of 1369-75, Charles V, King of France, won back from Edward what had been conceded in 1360. By 1375, when a truce was made at Bruges, English possessions in France had been reduced to Calais, a coastal strip of territory from Bordeaux to Boulogne, and parts of the Brittany coast.
↑ e.g. Knapton, S. (2014, December 02). Richard III DNA shows British Royal family may not have royal bloodline. Telegraph. www.telegraph.co.uk (Note Elizabeth of York's claim was through Lionel of Antwerp, not John of Gaunt.)
It may be possible to confirm family relationships with Edward III by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
Y-chromosome DNA test-takers in his direct paternal line on WikiTree:
According to Wikitree relationship finder, Edward was my 20th gr. grandfather through the Radcliffe’s, the Canbys (Quaker, Bucks Co PA),and the Gillingham's (Quaker, PA). My grandmother was a Gillingham.
I am related to Edward Plantagenet of England and Eleanor Plantagenet, Queen of France through my Williams, Morgan/Cromwell/Tudors. My DNA is matched to Roger Williams, b. 1700's, founder of Rhode Island, and his family goes back to Morgan Williams/Cromwell as well as the Percy family, Dukes of Northumberland. My daughter's DNA, Sheryl (Sanders) Bolton's DNA is matched to the Bourbons, Capet, Sanders Family. O am now working on the Dukes of Norhumberland, Percys.