Reference about the Banach-Tarski paradox I think the title says it all. I am planning on giving a talk in a few weeks about the Banach-Tarski paradox and I have some pdfs found online which describe the paradox a little but I am looking for a solid reference which covers the construction from A to Z and on which I can extract the main ideas for my talk from (I understand the ideas beneath the paradox, I am just looking for a formal proof with no details excluded,i.e. a well-structured document). Anyone has a reference in mind? 
 A: In my view the images that show the constructive version of the BTP in hyperbolic space are the best to motivate what is going on. Sure, there are some details that are different in $R^3$ compared to $H^2$, but they are just details really. The underlying group theory -- the way a free group leads to a paradox -- is so clear in the hyperbolic plane, disk model. Of course, this is discussed in my BTP book, but this demo/movie (which anyone can look at with the free software) shows the story nicely.
Stan Wagon
A: I gave a similar presentation at MathFest 2011 in Kentucky last week, using Stan Wagon's book as a guide.  Here is a list of definitions/theorems/etc that are on the direct thread to reaching the Banach-Tarski Paradox (stated as a corollary in the book).


*

*Def 1.1: G-Paradoxical

*Thm 1.2: Free group of Rank 2 F-paradoxical

*Prop 1.10: Group acting without nontrivial fixed points

*Thm 2.1: $SO_3$ has free subgroup of Rank 2 

*Thm 2.3: Hausdorff Paradox 

*Def 3.3: G-Equidecomposable 

*Prop 3.4: Equidecomposability preserves Paradoxes 

*Thm 3.9: $S^2$ and $S^2$ minus a countable set are equidecomposable 

*Cor 3.10: The Banach-Tarski Paradox


As others have stated, there are MANY interesting results along the way in this book, and the development is superb.  Here is a link to a modified version of the presentation I gave at MathFest.  It is an attempt at illustrating exactly what is presented in the material of the text, rather than providing an alternative interpretation (baby steps, right?).  For the web version, I've added some annotations so that it's better suited for reading as the original slide presentation didn't have a lot of textual development, though unfortunately I was not able to add detailed descriptions of the animations without significant re-work.
A: The Banach-Tarski Paradox, a great book by Stan Wagon, quite detailed. Most university libraries would have it.  The book also discusses a lot of interesting ancillary material, very useful for a lecture!
Comment: The result does not extend to $\mathbb{R}^2$.  Roughly speaking, this is because there is a (finitely additive) translation invariant "measure" on all subsets of $\mathbb{R}^2$  that extends Lebesgue measure.  
The following is an old question of Tarski: Given a disc and a square of equal area, can the disc be decomposed into a finite number of regions, which can be reassembled to form the square?  About $20$ years ago, Laczkovich proved, to everyone's surprise, that the answer is yes.
