In this paper, we present a new two-dimensional viscometer, and the hydrodynamic calculations used to obtain the surface viscosities from the measurements. In order to interpret the experiments, performed with solutions of sodium dodecyl sulfate (SDS) and also with monolayers of insoluble surfactants, we develop various hydrodynamic models of soluble Gibbs monolayers and of incompressible Langmuir monolayers, that describe well the experimental results. In the case of SDS solutions, the calculations allow the determination of the surface shear viscosity, and its value is in good agreement with previous studies.