  We announce results about flat (linkless) embeddings of graphs in 3-space. A
piecewise-linear embedding of a graph in 3-space is called {\it flat} if every
circuit of the graph bounds a disk disjoint from the rest of the graph. We have
shown:
  (i) An embedding is flat if and only if the fundamental group of the
complement in 3-space of the embedding of every subgraph is free.
  (ii) If two flat embeddings of the same graph are not ambient isotopic, then
they differ on a subdivision of $K_5$ or $K_{3,3}$.
  (iii) Any flat embedding of a graph can be transformed to any other flat
embedding of the same graph by ``3-switches'', an analog of 2-switches from the
theory of planar embeddings. In particular, any two flat embeddings of a
4-connected graph are either ambient isotopic, or one is ambient isotopic to a
mirror image of the other.
  (iv) A graph has a flat embedding if and only if it has no minor isomorphic
to one of seven specified graphs. These are the graphs that can be obtained
from $K_6$ by means of $Y\Delta$- and $\Delta Y$-exchanges.
