This paper describes the workspace and the inverse and direct kinematic analysis of the VERNE machine, a serial/parallel 5-axis machine tool designed by Fatronik for IRCCyN. This machine is composed of a three-degree-of-freedom (DOF) parallel module and a two-DOF serial tilting table. The parallel module consists of a moving platform that is connected to a fixed base by three non-identical legs. This feature involves (i) a simultaneous combination of rotation and translation for the moving platform, which is balanced by the tilting table and (ii) workspace whose shape and volume vary as a function of the tool length. This paper summarizes results obtained in the context of the European projects NEXT ("Next Generation of Productions Systems").