  If D is a rational polygon, then the associated rational billiard table is
given by \Omega(D). Such a billiard table is well understood. If F is a closed
fractal curve approximated by a sequence of rational polygons, then the
corresponding fractal billiard table is denoted by \Omega(F). In this paper, we
survey many of the results from [LapNie1-3] for the Koch snowflake fractal
billiard \Omega(KS) and announce new results on two other fractal billiard
tables, namely, the T-fractal billiard table \Omega(T) (see [LapNie6]) and a
self-similar Sierpinski carpet billiard table \Omega(S_a) (see [CheNie]).
  We build a general framework within which to analyze what we call a sequence
of compatible orbits. Properties of particular sequences of compatible orbits
are discussed for each prefractal billiard \Omega(KS_n), \Omega(T_n) and
\Omega(S_a,n), for n = 0, 1, 2... . In each case, we are able to determine a
particular limiting behavior for an appropriately formulated sequence of
compatible orbits. Such a limit either constitutes what we call a nontrivial
path of a fractal billiard table \Omega(F) or else a periodic orbit of
\Omega(F) with finite period. In our examples, F will be either KS, T or S_a.
Several of the results and examples discussed in this paper are presented for
the first time.
  We then close with a brief discussion of open problems and directions for
further research in the emerging field of fractal billiards.
