1 Introduction

2 The quantum Teichmüller space: foundation
2.1 Ideal triangulations
2.2 Shear coordinates for the Teichmüller space
2.3 The Chekhov-Fock algebra
2.4 The quantum Teichmüller space

3 The quantum Teichmüller space: properties
3.1 Uniqueness
3.2 Representations

4 Kashaev algebra
4.1 Decorated ideal triangulations
4.2 Kashaev coordinates
4.3 Generalized Kashaev algebra: triangulation-dependent
4.4 Generalized Kashaev algebra: triangulation-independent

5 Kashaev coordinates and shear coordinates
5.1 Decorated ideal triangulations
5.2 Space of Kashaev coordinates
5.3 Exact sequence
5.4 Relation to bivectors
5.5 Compatibility of coordinate changes

6 Relationship between quantum Teichmüller space and Kashaev algebra
6.1 Homomorphism
6.2 Compatibility
6.3 Quotient algebra