Euler equations describing the potential flow of an incompressible ideal fluid with free surface in 1+1 dimensions can be studied efficiently by performing the conformal mapping to the lower half-plane. For the finite depth fluid the half-plane has to be replaced by a horizontal strip. The Euler equations in conformal variables are Hamiltonian, though the both symplectic and Poisson structures are not canonical. In essential degree, the surface dynamics is defined by motion of singularities of the Jacobian for the conformal mapping posed in the upper half-plane. The Jacobian’s zeros generate commuting motion constants of the system. It does not mean that the system becomes integrable. Integrability takes place only in two limiting cases: when Jacobian has a ”narrow” cut and when the surface is ”almost flat”. In these cases the system is reduced to the complex one, then to the real Hopf equation. In a general case, the cut is a structurally stable type of singularity. The equations imposed on the spectral densities on a cut could be written in the closed form. Equations of free-surface hydrodynamics in conformal variables are convenient for numerical simulation. To solve them, one can use either a spectral code or a direct numerical calculation of the Hilbert transform. Using the conformal variables we performed the numerical simulation of a nonlinear stage of the Stokes wave modulational instability. We showed that the development of the instability for waves of a moderate amplitude (ka ' 0.15) leads to formation of sporadic freak waves with the amplitude exceeding the Stokes limit (ka ' 0.5). This leads us to the hypothesis that freak waves are a result of modulational instability.