  This will is an expository description of quadratic rational maps. Sections 2
through 6 are concerned with the geometry and topology of such maps. Sections
7--10 survey of some topics from the dynamics of quadratic rational maps. There
are few proofs. Section 9 attempts to explore and picture moduli space by means
of complex one-dimensional slices. Section 10 describes the theory of real
quadratic rational maps. For convenience in exposition, some technical details
have been relegated to appendices: Appendix A outlines some classical algebra.
Appendix B describes the topology of the space of rational maps of degree
\[d\]. Appendix C outlines several convenient normal forms for quadratic
rational maps, and computes relations between various invariants.\break
Appendix D describes some geometry associated with the curves
\[\Per_n(\mu)\subset\M\]. Appendix E describes totally disconnected Julia sets
containing no critical points. Finally, Appendix F, written in collaboration
with Tan Lei, describes an example of a connected quadratic Julia set for which
no two components of the complement have a common boundary point.
