Two of the 41 signers of the Mayflower Compact were Fullers, brothers Samuel and Edward, and both came over on the Mayflower from Leiden, Holland, in 1620.
Samuel was an educated and well-respected man. He was not only a surgeon (physician) to Plymouth Colony, but also apparently a man with some relgious learning, although not a minister himself. He was consulted in 1629 about the founding of Salem Church, and next year assisted also in the founding of the Massachusetts Bay Colony. He continued to practice medicine until his death in 1633, and in his will he bequeathed to Roger Williams, who was soon to be minister of Salem Church, one of his books on "physic". Samuel left a son Samuel, from whom there are many Fuller descendants. Although I do not have descent from Samuel, Robert Charles Anderson tells us that apparently he made such an impression on the early settlers of Massachusetts Bay that several of them placed their children in his household, presumably for their social and educational improvement, and among these was an ancestor, Elizabeth Cole, daughter of Rice.
"Edward Fuller and his wife died soon after they came ashore" says Governor Bradford in his journal, however "their son Samuel is living and married and hath four children or more." Samuel, the son, was brought up by his Uncle Samuel, but when he came of age seems to have removed from central Plymouth to Barnstable and to have lived a quiet life there. However, he did mary Jane Lothrop, daughter of one of the first ministers of Plymouth, and sired several sons.
Edward also had an older son, Matthew Fuller, who, it appears, came over much later, because the first record for him in the colony is dated 26Oct1640, by which time he was already married and had several children. Matthew followed in his Uncle Samuel's footsteps as a surgeon, and he also had a military bent, serving as Capt. Miles Standish's lieutenant in 1654, and later as Captain of the Plymouth forces in his own right. During the period of King Philip's War, he served on the United Colonies Council of War, and also as Surgeon General of the colonial forces against the Indians.
I have descents from both of Edward's children, Samuel, and Captain Matthew, through 5th generation Mayflower descendant, Roger Fuller, of Hebron, Connecticut.
Most of the Mayflower lines are very thoroughly worked out by now, and the definitive and most up-to-date reference on the first generation of Plymouth colonists is Robert Charles Anderson, The Pilgrim Migration (NEHGS, 2004), while for those interested in learning whether the descend from one of the Plymouth Fullers, the two volumes of the Mayflower Families Through Five Generations series, published by The Mayflower Society (Volume 1: Francis Eaton, Samuel Fuller, and William White; and Volume 4: Edward Fuller) are the works to consult.