To realise the miniaturised devices, the precise measurement of nanoscale tunnelling current in ultrathin films is of utmost importance. For the nanoscale current measurements, current sensing atomic force microscope (CSAFM) is one of the most powerful tool. CSAFM allows to map the current distribution on the film surface and it permits to perform current measurements as a function of applied bias voltage. It has turned out to be crucial for studies of organic films. In CSAFM, a physical contact is made on film with a precise control of the applied force in nanonewton (nN) range. For the preparation of ultrathin film, Langmuir-Blodgett (LB) technique is known to provide a uniform film with a good control over the thickness in the molecular level. In the last two decades, there have been many CSAFM studies for the tunnelling current measurements. This review is intended to cover the literature on the tunnelling current measurements using CSAFM.