From Caleb Johnson's Mayflower History Website, http://www.mayflowerhistory.com:
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 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).
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 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.
From Mayflower Families Through Five Generations, Vol. 18 Part 1 Third Edition.
In addition, we have found that the wife of Richard Warren, has been indentified as Elizabeth WALKER, daughter of Augustine Walker of Great Amwell, Hertforshire. In the April 2003 issue of The American Genealogist (TAG) Vol. 78, No. 2 (Whole Number 310), an article entitled, "The Marriage of Richard Warren of the Mayflower" by Edward J. Davies, presents evidence for this identification of the maiden name of Elizabeth. This evidence is accepted by leading researchers in the field as being excellent and accurate and overriding any previous identifications.
Judith H. Swan
RICHARD1 WARREN, b. prob. England; d. Plymouth 1628.
He m. prob. England prior to 1610 ELIZABETH WALKER, b. ca. 1580; d. Plymouth 2 Oct. 1673 aged above 90 years.
Governor Bradford wrote "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."
The wife and five daughters came to Plymouth on the Anne in 1623.
In "Mourt's Relation" under date of "sixt of December" it is stated "and three of london, Richard Warren, Steeven Hopkins and Edward Dotte ..." This statement that he was from London is all we know about the origin of Richard Warren despite considerable research to learn more. (See MQ 51:109112.)
The 22 May 1627 Division of Cattle names Richard Warren, wife Elisabeth Warren, Nathaniell Warren, Joseph Warren, Mary Warren, Anna Warren, Sara Warren, Elizabeth Warren and Abigail Warren.
In a deed dated 28 Sept 1629 the land which Thomas Clarke sold to William Bradford was bounded on one side by the land of "widow Warren."
At the 7 March 1636/7 court it was agreed that Elizabeth Warren, widow, the relict of Mr. Richard Warren, deceased, was to be a "Purchaser" as she had performed the bargain after her husband's decease, and also for confirming the land formerly given by her to her sons-in-law Richard Church, Robert Bartlett and Thomas Little in marriage with their wives, her daughters.
In a codicil to his will dated 16 July 1667 Nathaniel Warren mentions his mother Elisabeth Warren, his brother Joseph Warren and his sisters Mary Bartlett Sr., Anna Little, Sarah Cooke, Elizabeth Church and Abigail Snow.
On 4 March 1673/4 Mary Bartlett, the wife of Robert Bartlett, ack she had received full satisfaction for her share of the estate of Mistris Elisabeth Warren, deceased; and John Cooke in behalf of all her sisters testified to the same. The court settled the remainder of the estate on Joseph Warren.
Children of Richard and Elisabeth (WALKER) Warren, first five b. prob. England, last two b. Plymouth:
2 i MARY2 WARREN, b. ca. 1610 (based on age at d.)
3 ii ANNA WARREN, b. ca. 1612; (based on age in deposition)
4 iii SARAH WARREN, b. ca. 1614
5 iv ELIZABETH WARREN, b. ca. 1616
6 v ABIGAIL WARREN, b. ca. 1618
7 vi NATHANIEL WARREN, b. ca. 1624 (based on age in deposition)
8 vii JOSEPH WARREN, b. ca. 1626
References: MD 1:14(Bradford's acct); 2:39(Nathaniel Warren cod.) 178-9(depos. of Anna, Nathaniel); 3:45-51 (Warren Notes). Plymouth Colony Recs 1:54(Eliz. made Purchaser); 2:177(Purchaser); 5:139-140(est. Elizabeth); 8:35(d. Eliz.); 12:12(1627 div.). New England's Memorial p. 135(d. Richard). Plymouth Colony LR 1:32(Eliz. called widow). Great Migration Begins 3:1935-7.