Richard Warren sailed from Plymouth, Eng., in the Mayflower, 6 September 1620. He was not of the Leyden Company, but joined the Pilgrims from London, and he was one of the signers of the Compact framed in the cabin of the Mayflower while in Cape Cod Harbor.  
Caleb Johnson's biosketch of Richard Warren states :
Richard Warren's English origins and ancestry have been the subject of much speculation, and countless different ancestries have been published for him, without a shred of evidence to support them. Luckily in December 2002, Edward Davies discovered the missing piece of the puzzle. Researchers had long known of the marriage of Richard Warren to Elizabeth Walker on 14 April 1610 at Great Amwell, Hertford. Since we know the Mayflower passenger had a wife named Elizabeth, and a first child born about 1610, this was a promising record. But no children were found for this couple in the parish registers, and no further evidence beyond the names and timing, until the will of Augustine Walker was discovered in December 2002 by Edward Davies. In the will of Augustine Walker, dated April 1613, he mentions "my daughter Elizabeth Warren wife of Richard Warren", and "her three children Mary, Ann and Sarah." We know that the Mayflower passenger's first three children were named Mary, Ann, and Sarah (in that birth order), and that they were born c1610, c1612, and c1614, so this put the nail in the coffin and we can say with near certainty that Richard Warren of the Mayflower married in Great Amwell, Hertford to Elizabeth Walker, daughter of Augustine Walker. Additional research is currently being sponsored by MayflowerHistory.com to see if anything further can be learned about these families.
Very little is known about Richard Warren's life in America. He came alone on the Mayflower in 1620, leaving behind his wife and five daughters. They came to him on the ship Anne in 1623, and Richard and Elizabeth subsequently had sons Nathaniel and Joseph at Plymouth. He received his acres in the Division of Land in 1623, and his family shared in the 1627 Division of Cattle. But he died a year later in 1628, the only record of his death being found in Nathaniel Morton's 1669 book New England's Memorial, in which he writes: "This year  died Mr. Richard Warren, who was an useful instrument and during his life bare a deep share in the difficulties and troubles of the first settlement of the Plantation of New Plymouth."
All of Richard Warren's children survived to adulthood, married, and had large families: making Richard Warren one of the most common Mayflower passengers to be descended from. Richard Warren's descendants include such notables as Civil War General and President Ulysses S. Grant, President Franklin D. Roosevelt, and Alan B. Shepard, Jr. the first American in space and the fifth person to walk on the moon. Their lineages are:
Ulysses S. Grant - Jesse Grant - Noah Grant - Susanna Delano - Jonathan Delano - Mercy Warren - Nathaniel Warren - Richard Warren
Franklin D. Roosevelt - Sara Delano - Warren Delano - Warren Delano - Ephraim Delano - Thomas Delano - Mercy Warren - Nathaniel Warren - Richard Warren
Alan B. Shepard, Jr. - Alan B. Shepard - Frederick Shepard - Rosina Johnson - Abner Johnson - Anna Delano - Silvanus Delano - Jonathan Delano - Mercy Warren - Nathaniel Warren - Richard Warren
The Great Migration Begins states, :
Migration: 1620 on Mayflower
First Residence: Plymouth
Estate: In the 1623 Plymouth division of land Richard Warren received an uncertain number of acres (perhaps two) as a passenger on the Mayflower, and five acres as a passenger on the Anne (presumably for his wife and children) [ PCR 12:4-6]. In the 1627 Plymouth division of cattle Richard Warren, his wife Elizabeth Warren, Nathaniel Warren, Joseph Warren, Mary Warren, Anna Warren, Sarah Warren, Elizabeth Warren and Abigail Warren were the first nine persons in the ninth company [ PCR 12:12]. He was one of the purchasers [ PCR 2:177].
In the 25 March 1633 Plymouth tax list Widow Warren was assessed 12s., and in the list of 27 March 1634, 9s. [ PCR 1:10, 27].
On 1 July 1633 "Mrs. Warren and Robt. Bartlet" were allowed to mow where they did the previous year, and again 14 March 1635/6 [ PCR 1:15, 41].
On 28 October 1633, a grant of Richard Warren's land on which he was required to erect a dwelling, returned to the court "for want of building" and it was regranted to Mr. Ralph Fogg, provided he pay Widow Warren sufficiently for her fence remaining there [ PCR 1:18].
On 7 March 1636/7 "it is agreed upon, by the consent of the whole Court, that Elizabeth Warren, widow, the relict of Mr. Richard Warren, deceased, shall be entered, and stand, and be purchaser instead of her said husband, as well because that (he dying before he had performed the said bargain) the said Elizabeth performed the same after his decease, as also for the establishing of the lots of lands given formerly by her unto her sons-in-law Richard Church, Robert Bartlett and Thomas Little, in marriage with their wives, her daughters" [ PCR 1:54, 2:177].
On 5 May 1640 "Richard Church, Rob[er]te Bartlett, Thomas Little, & Mrs. Elizabeth Warren are granted enlargements at the heads of their lots to the foot of the Pyne Hills, leaving a way betwixt them and the Pyne Hills, for cattle and carts to pass" [ PCR 1:152].
On 11 June 1653, as the result of a disagreement between Mrs. Elizabeth Warren and her son, Nathaniel, and a petition offered in court by Mrs. Jane Collier on behalf of her grandchild, Sarah, wife of Nathaniel Warren, the court chose four indifferent men to settle the matter of access to lands [ MD 2:64, citing PCLR 2:73].
On 4 March 1673/4 Mary Bartlett, wife of Robert Bartlett, came into this court and owned "that she hath received full satisfaction for whatsoever she might claim as due from the estate of Mistris Elizabeth Warren, deceased, and John Cooke, in the behalf of all her sisters, testified the same before the court; and the court doth hereby settle the remainder of the said estate on Joseph Warren" [ PCR 5:139-40].
Birth: By about 1578 based on estimated date of marriage.
Death: Plymouth 1628. ("This year died Mr. Richard Warren, who hath been mentioned before in this book, and was an useful instrument; and during his life bore a deep share in the difficulties and troubles of the first settlement of the plantation of New-Plymouth" [ Morton 85].
Marriage: By about 1609 Elizabeth _____; she died at Plymouth on 2 October 1673, aged about 90 (probably an exaggeration) [ PCR 8:35]. (Elizabeth's maiden name has been given as "March" in many sources, without documentation).
i Mary, born say 1609; m. say 1629 Robert Bartlett (date based on estimated age of children at their marriages).
ii Ann, born say 1613; m. Plymouth 19 April 1633 Thomas Little [ PCR 1:13].
iii Sarah, born say 1614; m. Plymouth 28 March 1634 John Cooke Junior [ PCR 1:29], son of Francis Cooke .
iv Elizabeth, born say 1615; m. by 1635/6 Richard CHURCH (he shared mowing land with Mrs. Warren 14 March 1635/6 [ PCR 1:41]).
v Abigail, born say 1619; m. Plymouth 8 (or 9) November 1639 Anthony Snow [ PCR 1:134].
vi Nathaniel, born Plymouth say 1624 (Bradford says he was born here, and his mother was a passenger on the Anne in 1623); m. Plymouth 19 November 1645 Sarah Walker [ PCR 2:94]. (See William Collier for discussion of her possible ancestry.)
vii Joseph, born Plymouth by 1627; m. about 1653 Priscilla Faunce, daughter of John Faunce (eldest child born Plymouth 23 September 1653 [ PCR 8:33]).
Comments: In his accounting of the passengers on the Mayflower Bradford included "Mr. Richard Warren, but his wife and children were left behind and came afterwards" [Bradford 442]. As of 1651, Bradford reported that "Mr. Richard Warren lived some four or five years and had his wife come over to him, by whom he had two sons before [he] died, and one of them is married and hath two children. So his increase is four. But he had five daughters more came over with his wife, who are all married and living, and have many children [Bradford 445-46].
Banks argued that Bradford's language in the sentence above meant that Richard Warren had two wives, with the first of whom he had five daughters and with the second of whom, Elizabeth, he had two sons [English Homes 92-93], and deForest agreed with him [Moore Anc 562].
Many attempts, all fruitless, have been made to discover the English origin of Richard Warren and the identity of his wife [MQ 51:109-12].
Richard Warren was in the party that explored the outer cape in early December 1620; he was described as being of London [Mourt 32].
On 5 July 1635, Thomas Williams, servant of widow Warren, confessed that "there being some dissention between him and his dame, she, after other things, exhorted him to fear God & do his duty, he answered, he neither feared God, nor the devil" [PCR 1:35]. He was reproved and released [ PCR 1:35].
On 5 January 1635/6 widow Warren paid 30s. to Thomas Clarke for borrowing his boat, and although returning it to a place of usual safety, an extraordinary storm wrecked it [PCR 1:36]. On 3 June 1639 "Mr. Andrew Hellot" was ordered to pay Mrs. Warren 10s. to settle an account between them [PCR 7:12].
Bibliographic Note: In 1938 L. Effingham deForest published a thorough study of Richard Warren [ Moore Anc 561-70]. Robert S. Wakefield, Janice A. Beebe and others have prepared the Richard Warren volume in the General Society of Mayflower Descendants' series of "pink books," the fifth edition of which was published in 1995 [ MFIP Warren].
Elizabeth Walker followed husband Richard to Plymouth, Massachusetts, in the Anne in 1623, accompanied by their daughters: Mary, Elizabeth, Ann, Sarah, and Abigail. Early Plymouth records suggest that Elizabeth was a woman of force and social position in the community. She was rated in the Plymouth tax list of 1632-3, and was one of the first purchasers of Dartmouth. She is usually spoken of as Mistress Elizabeth Warren, a designation by no means common, and she is one of the rare instances in that early colony of continued widowhood.
"Mistris Elizabeth Warren an aged widdow aged above 90 yeares Deceased on the [twenty-] second of October 1673 whoe haveing lived a Godly life Cam to her Grave as a shoke of Corn fully ripe shee was honoralby buried on the 24th of October aforsaid." "Plymouth Colony vital records," 
For more information on Elizabeth, see 
 Mrs. Washington A. Roebling, Richard Warren of the Mayflower, and some of his descendants (Boston: 1901), 3, [HathiTrust].
 New England Historical and Genealogical Register 55 (1901), at 70-71, [InternetArchive].
 Caleb H. Johnson, MayflowerHistory.com, [Caleb_Johnson_Mayflower_History].
 Robert Charles Anderson, The Great Migration Begins: Immigrants to New England, 1620-1633 (Boston, Massachusetts: New England Historic Genealogical Society, 1995), 1935, [AmericanAncestors].
 Mayflower Descendant, Transcribed by George Ernest Bowman, various volumes.
 Edward J. Davies, "The Marriage of Richard Warren of the Mayflower," The American Genealogist 78 (2003), 81-86, at 81, [AmericanAncestors].