What is a good introduction to integrable models in physics? I would be interested in a good mathematician-friendly introduction to integrable models in physics, either a book or expository article.
Related MathOverflow question: what-is-an-integrable-system.
 A: My references are very good reviews: 
Quantum inverse scattering and Algebraic Bethe Ansatz:
Faddeev: How Algebraic Bethe Ansatz works for integrable model
Kulish and Sklyanin: Quantum Spectral Transform Method. Recent Developments
Takhtajan: Introduction to algebraic Bethe ansatz
and the Books: 
Jimbo and Miwa: Algebraic Analysis of Solvable Lattice Models
Korepin et al: Quantum inverse Scattering and Correlation Functions
Korepin et al: The One-Dimensional Hubbard Model 
plus the article 
Martins and Ramos: The Quantum Inverse Scattering Method for Hubbard-like Models
A: I take "integrable models" to mean "exactly solvable models in statistical physics".
You can take a look at the classic book


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*R. J. Baxter - Exactly Solved Models in Statistical mechanics (You can download it for free)


Otherwise this new book is quit readable and covers more than just solvable models


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*G. Mussardo - Statistical field theory: an introduction to exactly solved models in statistical physics
Others can probably give you more mathematician-friendly references, but I think it would be good if you could be more specific about what you are looking for.
A: 
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*Another good recent book: Maciej Dunajski, Solitons, Instantons and Twistors.

*Some very good reviews on Quantum inverse scattering and Algebraic Bethe Ansatz:


*

*Faddeev: How Algebraic Bethe Ansatz works for integrable model

*Kulish and Sklyanin: Quantum Spectral Transform Method. Recent Developments

*Takhtajan: Introduction to algebraic Bethe Ansatz
and the Books: 


*

*Jimbo and Miwa: Algebraic Analysis of Solvable Lattice Models

*Korepin et al: Quantum inverse Scattering and Correlation Functions

*Korepin et al: The One-Dimensional Hubbard Model
plus the article 


*

*Martins and Ramos: The Quantum Inverse Scattering Method for Hubbard-like Models



Here are some more


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*The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension

*Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems

*Exactly Solved Models: A Journey in Statistical Mechanics : Selected Papers with Commentaries

*Introduction to Classical Integrable Systems

*Classical Many-Body Problems Amenable to Exact Treatments

*Exact Methods in Low-dimensional Statistical Physics and Quantum Computing
