This paper provides numerical and experimental results for the first-, second- and third-order hydrodynamic forces acting upon cylinders oscillating vertically with large amplitude on a free surface. The procedure of calculations is based on the theory in which it is assumed that the thickness of the cylinder and the free surface elevation are small enough but the amplitude of the oscillation is so large that the second-order contribution to the forces could not be neglected. Numerical calculations are made for two elliptic cylinders, a circular cylinder, a stern-section cylinder, whose section is similar to the one of a ship at the stern, and a wedge, and their results are compared with those obtained through forced oscillation tests. It is observed that the calculated values for the second-order hydrodynamic forces, in particular, are in fairly good agreement with the measured ones.