It is well-known, and was first established by Knuth in 1969, that the number
of 321-avoiding permutations is equal to that of 132-avoiding permutations. In
the literature one can find many subsequent bijective proofs of this fact. It
turns out that some of the published bijections can easily be obtained from
others. In this paper we describe all bijections we were able to find in the
literature and show how they are related to each other via ``trivial''
bijections. We classify the bijections according to statistics preserved (from
a fixed, but large, set of statistics), obtaining substantial extensions of
known results. Thus, we give a comprehensive survey and a systematic analysis
of these bijections. We also give a recursive description of the algorithmic
bijection given by Richards in 1988 (combined with a bijection by Knuth from
1969). This bijection is equivalent to the celebrated bijection of Simion and
Schmidt (1985), as well as to the bijection given by Krattenthaler in 2001, and
it respects 11 statistics--the largest number of statistics any of the
bijections respects