Who were the Sons of God who seem to have begat the Nephilim? Genesis 6:1--4 tells about the "sons of God" marrying whomsoever they pleased, and the Nephilim---"the heroes of old, men of renown"---were begotten then. How is this to be interpreted?
Dr John Millam, on the Reasons to Believe website, has this to say on the Nephilim (about 1/2 way through, the section entitled The Sons of God/Giants):


  Historically, the most popular way of understanding this passage was
  that the "sons of God" refer to angels or to the descendents of the
  godly line of Seth, Adam's son. The translators of the Septuagint,
  however, biased interpretations toward one standpoint by substituting
  "angels" for "sons of God". (The sons-of-God-as-angels view was
  fashionable in Jewish circles at that time. It appears in the
  apocryphal Book of Enoch, so the translators of the Septuagint were
  simply reflecting popular thinking.) This helps explain why the angel
  interpretation held exclusive dominance in the church for the first
  two centuries. A second example can be found in this same passage
  where the Septuagint translated Nephilim (literally "fallen ones") as
  "giants" (Greek gigos). As a result, the view that the Nephilim were
  "giants" was rampant in the early church.


Most people I have read on the matter agree that it is highly uncertain who these "Nephilim" actually were, and as a result, speculation abounds.  In my thinking it seems problematic to consider that angels, who are purely spirit beings could mate and produce offspring with humans who are a distinct creation of body into which God breathed spirit.

From the wikipedia article:


  Some individuals and groups, including St. Augustine, John Chrysostom,
  and John Calvin, take the view of Genesis 6:2 that the "Angels" who
  fathered the Nephilim referred to certain human males from the lineage
  of Seth, who were called sons of God probably in reference to their
  being formerly in a covenantal relationship with Yahweh (cf.
  Deuteronomy 14:1; 32:5); according to these sources, these men had
  begun to pursue bodily interests, and so took wives of the daughters
  of men, e.g., those who were descended from Cain or from any people
  who did not worship God.


While the ca. 6th century The Conflict of Adam and Eve with Satan (a Christian pseudepigraphical work found in Ge'ez) says:


  Certain wise men of old wrote concerning them, and say in their
  [sacred] books, that angels came down from heaven, and mingled with
  the daughters of Cain, who bare unto them these giants. But these
  [wise men] err in what they say. God forbid such a thing, that angels
  who are spirits, should be found committing sin with human beings.
  Never, that cannot be. And if such a thing were of the nature of
  angels, or Satans, that fell, they would not leave one woman on earth,
  undefiled... But many men say, that angels came down from heaven, and
  joined themselves to women, and had children by them. This cannot be
  true. But they were children of Seth, who were of the children of
  Adam, that dwelt on the mountain, high up, while they preserved their
  virginity, their innocence and their glory like angels; and were then
  called 'angels of God.' But when they transgressed and mingled with
  the children of Cain, and begat children, ill-informed men said, that
  angels had come down from heaven, and mingled with the daughters of
  men, who bare them giants.


Which seems to make sense to me.  Both wikipedia articles contain numerous references which may be of interest for further research and reading.

A second reference to Nephilim occurs in Numbers 13:32-33, but since all of mankind was destroyed in the flood of Gen 9 ff, I don't see how they could be of the same bloodline.  So that suggests that Nephilim might be more of a general than specific descriptive term.  It may be significant that both groups of men are described as "giants" (note also that Goliath and his kin, "descendants of Rapha in Gath" were described as having six fingers and six toes, which indicates some sort of genetic deformity).