Let A be a local ring of dimension d. If A is a quotient of a regular local ring of dimension n = d+r, then we say that A hasembeddingcodimension≤ r. This paper investigates some special properties of local rings of small embedding codimension. The main idea is to exploit a result in [17], which says that local rings of small embedding codimension and depth ≥ 3 are parafactorial. This tells us, with suitable additional hypotheses, that the ring is factorial, or Gorenstein, or even a complete intersection.