The basic unit of information in filamentary-based resistive switching memories is physically stored in a conductive filament. Therefore, the overall performance of the device is indissolubly related to the properties of such filament. In this Letter, we report for the first time on the three-dimensional (3D) observation of the shape of the conductive filament. The observation of the filament is done in a nanoscale conductive-bridging device, which is programmed under real operative conditions. To obtain the 3D-information we developed a dedicated tomography technique based on conductive atomic force microscopy. The shape and size of the conductive filament are obtained in three-dimensions with nanometric resolution. The observed filament presents a conical shape with the narrow part close to the inert-electrode. On the basis of this shape, we conclude that the dynamic filament-growth is limited by the cation transport. In addition, we demonstrate the role of the programming current, which clearly influences the physical-volume of the induced conductive filaments.