  The following notes provide an introduction to recent work of Branner,
Hubbard and Yoccoz on the geometry of polynomial Julia sets. They are an
expanded version of lectures given in Stony Brook in Spring 1992. I am indebted
to help from the audience.
  Section 1 describes unpublished work by J.-C. Yoccoz on local connectivity of
quadratic Julia sets. It presents only the "easy" part of his work, in the
sense that it considers only non-renormalizable polynomials, and makes no
effort to describe the much more difficult arguments which are needed to deal
with local connectivity in parameter space. It is based on second hand sources,
namely Hubbard together with lectures by Branner and Douady. Hence the
presentation is surely quite different from that of Yoccoz.
  Section 2 describes the analogous arguments used by Branner and Hubbard to
study higher degree polynomials for which all but one of the critical orbits
escape to infinity. In this case, the associated Julia set J is never locally
connected. The basic problem is rather to decide when J is totally
disconnected. This Branner-Hubbard work came before Yoccoz, and its technical
details are not as difficult. However, in these notes their work is presented
simply as another application of the same geometric ideas.
  Chapter 3 complements the Yoccoz results by describing a family of examples,
due to Douady and Hubbard (unpublished), showing that an infinitely
renormalizable quadratic polynomial may have non-locally-connected Julia set.
An Appendix describes needed tools from complex analysis, including the
Gr\"otzsch inequality.
