The first two lectures are intended to be an overview, with brief sketches of proofs:

Lecture 1: Uniqeness for critically periodic folding maps of the interval

Lecture 2: Uniqueness and Existence for post-critically finite rational maps

Lecture 3: Extremal length and quadratic differentials

Lecture 4: Quadratic differentials and Teichmuller's theorem

Lecture 5: Teichmuller's Uniqueness Theorem

Lecture 6: Some of the ideas for Teichmuller's Existence Theorem

Lecture 7: The general definition of K-qc mappings

Lecture 8 is taken out of Chapter 2 of Ahlfors' Lectures on Quasiconformal Mappings

Lecture 9: The proof of Teichmuller's Existence Theorem

Lecture 10: The Teichmuller Space

Lecture 11: The Bers embedding

Lecture 12: Combinatorial Rigidity of Post-critically Finite Rational Maps

Lecture 13: Thurston's iteration on Teichmuller space

Lecture 14: Why Thurston's iteration has a fixed point or a combinatorial obstruction.