  This paper is an attempt to survey the current state of our knowledge on the
Caccetta-Haggkvist conjecture and related questions. In January 2006 there was
a workshop hosted by the American Institute of Mathematics in Palo Alto, on the
Caccetta-Haggkvist conjecture, and this paper partly originated there, as a
summary of the open problems and partial results presented at the workshop.
This summary includes results and open problems related to Caccetta-Haggkvist,
Seymour's Second Neighborhood Conjecture, the k/2 Conjecture (for nonedges),
and connections with algebraic number theory through Cayley graphs, along with
a number of other related topics.
