Tightly moored single-body floating devices are an important class of offshore wave energy converters. Examples are the devices under development at the University of Uppsala, Sweden, and Oregon State University, USA, prototypes of which were recently tested off the western coast of Sweden and off the Oregon coast, respectively. These devices are equipped with a linear electrical generator. The mooring system consists of a cable that is kept tight by a spring or equivalent device. This cable also prevents the buoy from drifting away by providing a horizontal restoring force. The horizontal and (to a lesser extent) the vertical restoring force are nonlinear functions of the vertical and horizontal displacements of the buoy, which makes the system a nonlinear one (even if the spring and damper are linear), whose modelling requires a time-domain analysis. Such an analysis is presented, preceded by a simpler frequency-domain approach. Numerical results (motions and absorbed power) are shown for a system consisting of a hemispherical buoy in regular and irregular waves, a tight mooring cable and a power take-off system consisting of a linear spring and a linear damper.