The aim of this paper is to provide the key theoretical and numerical aspects of the NEMOH software. NEMOH is a numerical solver for computations of first order hydrodynamic coefficients in frequency domain. It is based on linear free surface potential flow theory. It has been developed in Ecole Centrale de Nantes for 30 years. It has been released in open source in January 2014. This paper starts with recalling the assumptions and the resulting boundary value problem for the free surface flow around a body with an arbitrary condition for the normal velocity on the body surface. Then, it is recalled how the 3D flow problem can be transformed into the 2D problem of a source distribution on the body surface using Green's second identity and the appropriate Green function. Discretization of the mathematical problem using the constant panel method leads to a linear matrix problem whose coefficients are the influence coefficients. Depending on the distance between the source and field point, the influence coefficients are calculated analytically or approximately. Irregular frequencies are removed by adding constraints on the interior flow problem. Hydrodynamic coefficients for usual diffraction and radiation problems can be obtained by specifying to the solver the appropriate body conditions. The flexibility in specifying the body conditions allows dealing with complex structures, e.g deformable wave energy converters. Eventually, it is explained how far field coefficients and free surface elevation can be obtained from the solver outputs.