The two-dimensional problem of hydrodynamic interaction of the horizontal circular cylinders with a free-surface is investigated both analytically and numerically. The fully nonlinear initial boundary-value problem is described and a numerical solution for it is presented. The free motion of a cylinder rising towards, or moving away from the free-surface or moving horizontally is studied. The numerical calculations are compared with a simple analytic theory in which we take the low- and high-frequency limit of the added mass, and the constant added mass of the submerged cylinder in the coefficients of the equation of motion. Further numerical calculations of an initially displaced, spring-loaded cylinder undergoing slow motions are compared with a simple analytic theory in which we also take the low-frequency limit of the added mass of the submerged cylinder. The aim is to provide a useful approximate method for simulation of various offshore operations. Fully nonlinear calculations of the free-surface deformations of the initially calm water caused by forced constant velocity motion of a totally submerged circular cylinder are compared with small-time asymptotics due to Tyvand & Miloh (1995). Their analytic results, which are taken to third order5 when gravity terms first appear in the expansion, are in excellent agreement with the numerical calculations for small times, beyond which only the numerical method will give accurate results, valid until the free-surface breaks. The breaking of the surface as a result of vertical downward motion is further investigated with the aim of establishing when and how this happens, since the phenomena causes the breakdown of the numerical calculations. The free motion of a cylinder entering a free-surface, initially half-submerged in calm water and having specific gravity of 1.2 is also investigated. This motion is pursued beyond the complete submergence stage, giving rise to interesting free-surface deformations and body dynamics. This study is complemented by a further investigation involving impulsively started and forced constant motion of a cylinder entering a free-surface at various angles and Froude numbers, and is also taken beyond the complete engulfment stage. Hydrodynamic forces on the cylinder obtained for various angles at the same Froude number are compared. Also, the hydrodynamic forces for the motion in the same direction at various Froude numbers are compared.