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.
• 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.
• 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.