1 Krichever-Novikov algebras: basic definitions and structure theory -- 1.1 Current, vector field, and other Krichever-Novikov algebras -- 1.2 Meromorphic λ-forms and Krichever-Novikov duality -- 1.3 Krichever-Novikov bases -- 1.4 Almost-graded structure, triangle decompositions -- 1.5 Central extensions and 2-cohomology -- Virasoro-type algebras -- 1.6 Affine Krichever-Novikov, in particular Kac-Moody, algebras -- 1.7 Central extensions of the Lie algebra D1g -- 1.8 Local cocycles for sl(n) and gl(n) -- 2 Fermion representations and Sugawara construction -- 2.1 Admissible representations and holomorphic bundles -- 2.2 Holomorphic bundles in the Tyurin parametrization -- 2.3 Krichever-Novikov bases for holomorphic vector bundles -- 2.4 Fermion representations of affine algebras -- 2.5 Verma modules for affine algebras -- 2.6 Fermion representations of Virasoro-type algebras -- 2.7 Sugawara representation -- 2.8 Proof of the main theorems for the Sugawara construction -- 2.8.1 Main theorems in the form of relations with structure constants -- 2.8.2 End of the proof of the main theorems -- 3 Projective flat connections on the moduli space of punctured Riemann surfaces and the Knizhnik-Zamolodchikov equation -- 3.1 Virasoro-type algebras and moduli spaces of Riemann surfaces -- 3.2 Sheaf of conformal blocks and other sheaves on the moduli space M(1,0)g,N+1 -- 3.3 Differentiation of the Krichever-Novikov objects in modular variables -- 3.4 Projective flat connection and generalized Knizhnik-Zamolodchikov equation -- 3.5 Explicit form of the Knizhnik-Zamolodchikov equations for genus 0 and genus 1 -- 3.5.1 Explicit form of the equations for g = 0 -- 3.5.2 Explicit form of the equations for g = 1 -- 3.6 Appendix: the Krichever-Novikov base in the elliptic case -- 4 Lax operator algebras.

4.1 Lax operators and their Lie bracket -- 4.1.1 Lax operator algebras for gl(n) and sl(n) -- 4.1.2 Lax operator algebras for sv(n) -- 4.1.3 Lax operator algebras for sp(2n) -- 4.2 Almost-graded structure -- 4.3 Central extensions of Lax operator algebras: the construction -- 4.4 Uniqueness theorem -- 5 Lax equations on Riemann surfaces, and their hierarchies -- 5.1 M-operators -- 5.2 L-operators and Lax operator algebras from M-operators -- 5.3 g-valued Lax equations -- 5.4 Hierarchies of commuting flows -- 5.5 Symplectic structure -- 5.6 Hamiltonian theory -- 5.7 Examples: Calogero-Moser systems -- 6 Lax integrable systems and conformal field theory -- 6.1 Conformal field theory related to a Lax integrable system -- 6.2 From Lax operator algebra to commutative Krichever-Novikov algebra -- 6.3 The representation of AL -- 6.4 Sugawara representation -- 6.5 Conformal blocks and the Knizhnik-Zamolodchikov connection -- 6.6 The representation of the algebra of Hamiltonian vector fields and commuting Hamiltonians -- 6.7 Unitarity -- 6.8 Relation to geometric quantization and quantum integrable systems -- 6.9 Remark on the Seiberg-Witten theory -- Bibliography -- Notation -- Index.