Quantum-classical dualities in integrable systems

We describe several interrelations between classical and quantum integrable systems of different types. The first example is the so-called quantum-classical duality between the classical many-body systems and quantum spin chains. It allows to identify spectrum of quantum Hamiltonians with the velocities of classical particles. More precisely, the quantum spectrum is defined by intersection of two Lagrangian submanifolds in the classical phase space. The second example comes from half-quantum Lax pairs for classical integrable many-body systems. It allows to obtain another family of quantum spin chains – the so-called long-range spin chains of Haldane-Shastry type. Finally, we discuss relation between classical and quantum Painleve equations coming from considering the classical linear problem.